%   ORIGINAL: h4/canonical/spolynom__distinct_c5
% 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/bool/TRUTH: T
% Assm: h4/canonical/SPmult0: h4/canonical/SPmult = h4/canonical/_20_40ind__typecanonical8
% Assm: h4/canonical/SPplus0: h4/canonical/SPplus = h4/canonical/_20_40ind__typecanonical7
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/canonical/spolynom__repfns_c0: !a. h4/canonical/_20_40ind__typecanonical9 (h4/canonical/_20_40ind__typecanonical10 a) = a
% Assm: h4/canonical/hidden____20__40ind____typecanonical8____def: h4/canonical/_20_40ind__typecanonical8 = (\a0 a1. h4/canonical/_20_40ind__typecanonical9 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/canonical/_20_40ind__typecanonical10 a0) (h4/canonical/_20_40ind__typecanonical10 a1)))
% Assm: h4/canonical/hidden____20__40ind____typecanonical7____def: h4/canonical/_20_40ind__typecanonical7 = (\a0 a1. h4/canonical/_20_40ind__typecanonical9 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/canonical/_20_40ind__typecanonical10 a0) (h4/canonical/_20_40ind__typecanonical10 a1)))
% Assm: h4/canonical/spolynom__distinct_c1: !a1 a0 a. ~(h4/canonical/SPvar a = h4/canonical/SPplus a0 a1)
% Assm: h4/canonical/spolynom__distinct_c4: !a1 a0 a. ~(h4/canonical/SPconst a = h4/canonical/SPmult a0 a1)
% Assm: h4/canonical/spolynom__distinct_c3: !a1 a0 a. ~(h4/canonical/SPconst a = h4/canonical/SPplus a0 a1)
% Assm: h4/canonical/spolynom__distinct_c2: !a1 a0 a. ~(h4/canonical/SPvar a = h4/canonical/SPmult a0 a1)
% Assm: h4/canonical/spolynom__size__def_c2: !f a1 a0. h4/canonical/spolynom__size f (h4/canonical/SPplus a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/canonical/spolynom__size f a0) (h4/canonical/spolynom__size f a1))
% Assm: h4/canonical/spolynom__size__def_c3: !f a1 a0. h4/canonical/spolynom__size f (h4/canonical/SPmult a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/canonical/spolynom__size f a0) (h4/canonical/spolynom__size f a1))
% Assm: h4/canonical/spolynom__11_c3: !a1_27 a1 a0_27 a0. h4/canonical/SPmult a0 a1 = h4/canonical/SPmult a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/canonical/spolynom__11_c2: !a1_27 a1 a0_27 a0. h4/canonical/SPplus a0 a1 = h4/canonical/SPplus a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/canonical/datatype__spolynom: !spolynom. h4/bool/DATATYPE (spolynom h4/canonical/SPvar h4/canonical/SPconst h4/canonical/SPplus h4/canonical/SPmult)
% Assm: h4/canonical/SPconst0: h4/canonical/SPconst = h4/canonical/_20_40ind__typecanonical6
% Assm: h4/canonical/hidden____20__40ind____typecanonical6____def: h4/canonical/_20_40ind__typecanonical6 = (\a. h4/canonical/_20_40ind__typecanonical9 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/canonical/SPvar0: h4/canonical/SPvar = h4/canonical/_20_40ind__typecanonical5
% Assm: h4/canonical/hidden____20__40ind____typecanonical5____def: h4/canonical/_20_40ind__typecanonical5 = (\a. h4/canonical/_20_40ind__typecanonical9 ((\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/canonical/spolynom__repfns_c1: !r. (\a0_27. !_27spolynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spolynom_27 a0 /\ _27spolynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spolynom_27 a0 /\ _27spolynom_27 a1) ==> _27spolynom_27 a0_270) ==> _27spolynom_27 a0_27) r <=> h4/canonical/_20_40ind__typecanonical10 (h4/canonical/_20_40ind__typecanonical9 r) = r
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/canonical/spolynom__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27spolynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spolynom_27 a0 /\ _27spolynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spolynom_27 a0 /\ _27spolynom_27 a1) ==> _27spolynom_27 a0_270) ==> _27spolynom_27 a0_27) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/canonical/spolynom__case__def_c3: !f3 f2 f1 f a1 a0. h4/canonical/spolynom__CASE (h4/canonical/SPmult a0 a1) f f1 f2 f3 = f3 a0 a1
% Assm: h4/canonical/spolynom__case__def_c2: !f3 f2 f1 f a1 a0. h4/canonical/spolynom__CASE (h4/canonical/SPplus a0 a1) f f1 f2 f3 = f2 a0 a1
% Assm: h4/canonical/spolynom__distinct_c0: !a_27 a. ~(h4/canonical/SPvar a = h4/canonical/SPconst a_27)
% Assm: h4/canonical/spolynom__11_c0: !a_27 a. h4/canonical/SPvar a = h4/canonical/SPvar a_27 <=> a = a_27
% Assm: h4/canonical/spolynom__size__def_c0: !f a. h4/canonical/spolynom__size f (h4/canonical/SPvar a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% Assm: h4/canonical/spolynom__case__def_c1: !f3 f2 f1 f a. h4/canonical/spolynom__CASE (h4/canonical/SPconst a) f f1 f2 f3 = f1 a
% Assm: h4/canonical/spolynom__11_c1: !a_27 a. h4/canonical/SPconst a = h4/canonical/SPconst a_27 <=> a = a_27
% Assm: h4/canonical/spolynom__case__def_c0: !f3 f2 f1 f a. h4/canonical/spolynom__CASE (h4/canonical/SPvar a) f f1 f2 f3 = f a
% Assm: h4/canonical/spolynom__size__def_c1: !f a. h4/canonical/spolynom__size f (h4/canonical/SPconst a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (f a)
% Assm: h4/canonical/canonical__sum__distinct_c2: !a2 a1_27 a1 a0_27 a0. ~(h4/canonical/Cons__monom a0 a1 a2 = h4/canonical/Cons__varlist a0_27 a1_27)
% Assm: h4/sum/sum__distinct: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm: h4/canonical/canonical__sum__repfns_c0: !a. h4/canonical/_20_40ind__typecanonical3 (h4/canonical/_20_40ind__typecanonical4 a) = a
% Assm: h4/canonical/hidden____20__40ind____typecanonical2____def: h4/canonical/_20_40ind__typecanonical2 = (\a0 a1. h4/canonical/_20_40ind__typecanonical3 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a00) (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/canonical/_20_40ind__typecanonical4 a1)))
% Assm: h4/canonical/canonical__sum__repfns_c1: !r. (\a0_27. !_27canonical__sum_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 a2 /\ _27canonical__sum_27 a2) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a00) (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27canonical__sum_27 a1) ==> _27canonical__sum_27 a0_270) ==> _27canonical__sum_27 a0_27) r <=> h4/canonical/_20_40ind__typecanonical4 (h4/canonical/_20_40ind__typecanonical3 r) = r
% Assm: h4/canonical/Cons__monom0: h4/canonical/Cons__monom = h4/canonical/_20_40ind__typecanonical1
% Assm: h4/canonical/hidden____20__40ind____typecanonical0____def: h4/canonical/_20_40ind__typecanonical0 = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM))
% Assm: h4/canonical/Nil__monom0: h4/canonical/Nil__monom = h4/canonical/_20_40ind__typecanonical0
% Assm: h4/canonical/Cons__varlist0: h4/canonical/Cons__varlist = h4/canonical/_20_40ind__typecanonical2
% Assm: h4/canonical/hidden____20__40ind____typecanonical1____def: h4/canonical/_20_40ind__typecanonical1 = (\a0 a1 a2. h4/canonical/_20_40ind__typecanonical3 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 (h4/canonical/_20_40ind__typecanonical4 a2)))
% 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/bool/FALSITY: !t. F ==> t
% Assm: h4/combin/UPD11__SAME__BASE: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/combin/UPDATE__def: !b a. h4/combin/UPDATE a b = (\f c. h4/bool/COND (a = c) b (f c))
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/arithmetic/ODD0_c1: !n. h4/arithmetic/ODD (h4/num/SUC n) <=> ~h4/arithmetic/ODD n
% Assm: h4/quote/index__distinct_c1: !a. ~(h4/quote/Left__idx a = h4/quote/End__idx)
% Assm: h4/while/OWHILE__IND: !f P G. (!s. ~G s ==> P s s) /\ (!s1 s2. G s1 /\ P (f s1) s2 ==> P s1 s2) ==> (!s1 s2. h4/while/OWHILE G f s1 = h4/option/SOME s2 ==> P s1 s2)
% Assm: h4/quote/End__idx0: h4/quote/End__idx = h4/quote/_20_40ind__typequote2
% Assm: h4/quote/Right__idx0: h4/quote/Right__idx = h4/quote/_20_40ind__typequote1
% Assm: h4/quote/hidden____20__40ind____typequote2____def: h4/quote/_20_40ind__typequote2 = h4/quote/_20_40ind__typequote3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/quote/Left__idx0: h4/quote/Left__idx = h4/quote/_20_40ind__typequote0
% Assm: h4/quote/hidden____20__40ind____typequote1____def: h4/quote/_20_40ind__typequote1 = (\a. h4/quote/_20_40ind__typequote3 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/quote/_20_40ind__typequote4 a)))
% Assm: h4/quote/index__repfns_c1: !r. (\a0. !_27index_27. (!a00. (?a. a00 = (\a1. h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27index_27 a) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27index_27 a) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM) ==> _27index_27 a00) ==> _27index_27 a0) r <=> h4/quote/_20_40ind__typequote4 (h4/quote/_20_40ind__typequote3 r) = r
% Assm: h4/quote/index__repfns_c0: !a. h4/quote/_20_40ind__typequote3 (h4/quote/_20_40ind__typequote4 a) = a
% Assm: h4/quote/hidden____20__40ind____typequote0____def: h4/quote/_20_40ind__typequote0 = (\a. h4/quote/_20_40ind__typequote3 ((\a0. h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/quote/_20_40ind__typequote4 a)))
% Assm: h4/prim__rec/wellfounded__def: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. R (f (h4/num/SUC n)) (f n))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/while/LEAST__EXISTS__IMP: !p. (?n. p n) ==> p (h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (h4/while/LEAST p) ==> ~p n)
% Assm: h4/arithmetic/FUNPOW0_c1: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (f x)
% Assm: h4/while/OWHILE__def: !s f G. h4/while/OWHILE G f s = h4/bool/COND (?n. ~G (h4/arithmetic/FUNPOW f n s)) (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (\n. ~G (h4/arithmetic/FUNPOW f n s))) s)) h4/option/NONE
% Assm: h4/arithmetic/FUNPOW0_c0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/prim__rec/LESS__MONO: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/PULL__EXISTS_c0: !Q P. (?x. P x) ==> Q <=> (!x. P x ==> Q)
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/pred__set/CARD__BIGUNION__SAME__SIZED__SETS: !s n. h4/pred__set/FINITE s /\ (!e. h4/bool/IN e s ==> h4/pred__set/FINITE e /\ h4/pred__set/CARD e = n) /\ (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s /\ ~(e1 = e2) ==> h4/pred__set/DISJOINT e1 e2) ==> h4/pred__set/CARD (h4/pred__set/BIGUNION s) = h4/arithmetic/_2A (h4/pred__set/CARD s) n
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/FINITE__BIGUNION__EQ: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm: h4/pred__set/BIGUNION__INSERT: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm: h4/pred__set/BIGUNION__EMPTY: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/CARD__EMPTY: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm: h4/pred__set/CARD__INSERT: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/DISJOINT__DEF: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/CARD__UNION__EQN: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/CARD (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t)) (h4/pred__set/CARD (h4/pred__set/INTER s t))
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Goal: !a1_27 a1 a0_27 a0. ~(h4/canonical/SPplus a0 a1 = h4/canonical/SPmult a0_27 a1_27)
%   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_bools_TRUTH]: T
% Assm [h4s_canonicals_SPmult0]: h4/canonical/SPmult = h4/canonical/_20_40ind__typecanonical8
% Assm [h4s_canonicals_SPplus0]: h4/canonical/SPplus = h4/canonical/_20_40ind__typecanonical7
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_canonicals_spolynomu_u_repfnsu_c0]: !a. h4/canonical/_20_40ind__typecanonical9 (h4/canonical/_20_40ind__typecanonical10 a) = a
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical8u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/canonical/_20_40ind__typecanonical8 x) x' = h4/canonical/_20_40ind__typecanonical9 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical10 x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical10 x') _0))))
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical7u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/canonical/_20_40ind__typecanonical7 x) x' = h4/canonical/_20_40ind__typecanonical9 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical10 x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical10 x') _0))))
% Assm [h4s_canonicals_spolynomu_u_distinctu_c1]: !a1 a0 a. ~(happ h4/canonical/SPvar a = happ (happ h4/canonical/SPplus a0) a1)
% Assm [h4s_canonicals_spolynomu_u_distinctu_c4]: !a1 a0 a. ~(happ h4/canonical/SPconst a = happ (happ h4/canonical/SPmult a0) a1)
% Assm [h4s_canonicals_spolynomu_u_distinctu_c3]: !a1 a0 a. ~(happ h4/canonical/SPconst a = happ (happ h4/canonical/SPplus a0) a1)
% Assm [h4s_canonicals_spolynomu_u_distinctu_c2]: !a1 a0 a. ~(happ h4/canonical/SPvar a = happ (happ h4/canonical/SPmult a0) a1)
% Assm [h4s_canonicals_spolynomu_u_sizeu_u_defu_c2]: !f a1 a0. h4/canonical/spolynom__size f (happ (happ h4/canonical/SPplus a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/canonical/spolynom__size f a0) (h4/canonical/spolynom__size f a1))
% Assm [h4s_canonicals_spolynomu_u_sizeu_u_defu_c3]: !f a1 a0. h4/canonical/spolynom__size f (happ (happ h4/canonical/SPmult a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/canonical/spolynom__size f a0) (h4/canonical/spolynom__size f a1))
% Assm [h4s_canonicals_spolynomu_u_11u_c3]: !a1_27 a1 a0_27 a0. happ (happ h4/canonical/SPmult a0) a1 = happ (happ h4/canonical/SPmult a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_canonicals_spolynomu_u_11u_c2]: !a1_27 a1 a0_27 a0. happ (happ h4/canonical/SPplus a0) a1 = happ (happ h4/canonical/SPplus a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_canonicals_datatypeu_u_spolynom]: !spolynom. h4/bool/DATATYPE (happ (happ (happ (happ spolynom h4/canonical/SPvar) h4/canonical/SPconst) h4/canonical/SPplus) h4/canonical/SPmult)
% Assm [h4s_canonicals_SPconst0]: h4/canonical/SPconst = h4/canonical/_20_40ind__typecanonical6
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical6u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/canonical/_20_40ind__typecanonical6 x = h4/canonical/_20_40ind__typecanonical9 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB x) _0))
% Assm [h4s_canonicals_SPvar0]: h4/canonical/SPvar = h4/canonical/_20_40ind__typecanonical5
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical5u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/canonical/_20_40ind__typecanonical5 x = h4/canonical/_20_40ind__typecanonical9 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x h4/bool/ARB) _0))
% Assm [h4s_canonicals_spolynomu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27spolynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27spolynom_27 a0 /\ happ _27spolynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27spolynom_27 a0 /\ happ _27spolynom_27 a1) ==> happ _27spolynom_27 a0_270) ==> happ _27spolynom_27 r) <=> h4/canonical/_20_40ind__typecanonical10 (h4/canonical/_20_40ind__typecanonical9 r) = r)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_canonicals_spolynomu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27spolynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27spolynom_27 a0 /\ happ _27spolynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27spolynom_27 a0 /\ happ _27spolynom_27 a1) ==> happ _27spolynom_27 a0_270) ==> happ _27spolynom_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_canonicals_spolynomu_u_caseu_u_defu_c3]: !f3 f2 f1 f a1 a0. h4/canonical/spolynom__CASE (happ (happ h4/canonical/SPmult a0) a1) f f1 f2 f3 = happ (happ f3 a0) a1
% Assm [h4s_canonicals_spolynomu_u_caseu_u_defu_c2]: !f3 f2 f1 f a1 a0. h4/canonical/spolynom__CASE (happ (happ h4/canonical/SPplus a0) a1) f f1 f2 f3 = happ (happ f2 a0) a1
% Assm [h4s_canonicals_spolynomu_u_distinctu_c0]: !a_27 a. ~(happ h4/canonical/SPvar a = happ h4/canonical/SPconst a_27)
% Assm [h4s_canonicals_spolynomu_u_11u_c0]: !a_27 a. happ h4/canonical/SPvar a = happ h4/canonical/SPvar a_27 <=> a = a_27
% Assm [h4s_canonicals_spolynomu_u_sizeu_u_defu_c0]: !f a. h4/canonical/spolynom__size f (happ h4/canonical/SPvar a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% Assm [h4s_canonicals_spolynomu_u_caseu_u_defu_c1]: !f3 f2 f1 f a. h4/canonical/spolynom__CASE (happ h4/canonical/SPconst a) f f1 f2 f3 = happ f1 a
% Assm [h4s_canonicals_spolynomu_u_11u_c1]: !a_27 a. happ h4/canonical/SPconst a = happ h4/canonical/SPconst a_27 <=> a = a_27
% Assm [h4s_canonicals_spolynomu_u_caseu_u_defu_c0]: !f3 f2 f1 f a. h4/canonical/spolynom__CASE (happ h4/canonical/SPvar a) f f1 f2 f3 = happ f a
% Assm [h4s_canonicals_spolynomu_u_sizeu_u_defu_c1]: !f a. h4/canonical/spolynom__size f (happ h4/canonical/SPconst a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (happ f a)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_distinctu_c2]: !a2 a1_27 a1 a0_27 a0. ~(happ (happ (happ h4/canonical/Cons__monom a0) a1) a2 = happ (happ h4/canonical/Cons__varlist a0_27) a1_27)
% Assm [h4s_sums_sumu_u_distinct]: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_repfnsu_c0]: !a. h4/canonical/_20_40ind__typecanonical3 (h4/canonical/_20_40ind__typecanonical4 a) = a
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/canonical/_20_40ind__typecanonical2 x) x = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical4 x) _0)))
% Assm [h4s_canonicals_canonicalu_u_sumu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27canonical__sum_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0 \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a0 a1) (h4/ind__type/FCONS a2 _0) /\ happ _27canonical__sum_27 a2) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a0) (h4/ind__type/FCONS a1 _0) /\ happ _27canonical__sum_27 a1) ==> happ _27canonical__sum_27 a0_270) ==> happ _27canonical__sum_27 r) <=> h4/canonical/_20_40ind__typecanonical4 (h4/canonical/_20_40ind__typecanonical3 r) = r)
% Assm [h4s_canonicals_Consu_u_monom0]: h4/canonical/Cons__monom = h4/canonical/_20_40ind__typecanonical1
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/canonical/_20_40ind__typecanonical0 = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0)
% Assm [h4s_canonicals_Nilu_u_monom0]: h4/canonical/Nil__monom = h4/canonical/_20_40ind__typecanonical0
% Assm [h4s_canonicals_Consu_u_varlist0]: h4/canonical/Cons__varlist = h4/canonical/_20_40ind__typecanonical2
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x. happ (happ (happ h4/canonical/_20_40ind__typecanonical1 x) x) x = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C x x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical4 x) _0)))
% 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_bools_FALSITY]: !t. F ==> t
% Assm [h4s_combins_UPD11u_u_SAMEu_u_BASE]: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (h4/sum/INL x) = happ f x) /\ (!y. happ h (h4/sum/INR y) = happ g y)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_combins_UPDATEu_u_def]: !b a x x. ?v. (v <=> a = x) /\ happ (h4/combin/UPDATE a b x) x = h4/bool/COND v b (happ x x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_arithmetics_ODD0u_c1]: !n. h4/arithmetic/ODD (h4/num/SUC n) <=> ~h4/arithmetic/ODD n
% Assm [h4s_quotes_indexu_u_distinctu_c1]: !a. ~(happ h4/quote/Left__idx a = h4/quote/End__idx)
% Assm [h4s_whiles_OWHILEu_u_IND]: !f P G. (!s. ~happ G s ==> happ (happ P s) s) /\ (!s1 s2. happ G s1 /\ happ (happ P (happ f s1)) s2 ==> happ (happ P s1) s2) ==> (!s1 s2. h4/while/OWHILE G f s1 = h4/option/SOME s2 ==> happ (happ P s1) s2)
% Assm [h4s_quotes_Endu_u_idx0]: h4/quote/End__idx = h4/quote/_20_40ind__typequote2
% Assm [h4s_quotes_Rightu_u_idx0]: h4/quote/Right__idx = h4/quote/_20_40ind__typequote1
% Assm [h4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/quote/_20_40ind__typequote2 = h4/quote/_20_40ind__typequote3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0)
% Assm [h4s_quotes_Leftu_u_idx0]: h4/quote/Left__idx = h4/quote/_20_40ind__typequote0
% Assm [h4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/quote/_20_40ind__typequote1 x = h4/quote/_20_40ind__typequote3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS (h4/quote/_20_40ind__typequote4 x) _0)))
% Assm [h4s_quotes_indexu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27index_27. (!a00. (?a. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (h4/ind__type/FCONS a _0) /\ happ _27index_27 a) \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a _0) /\ happ _27index_27 a) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0 ==> happ _27index_27 a00) ==> happ _27index_27 r) <=> h4/quote/_20_40ind__typequote4 (h4/quote/_20_40ind__typequote3 r) = r)
% Assm [h4s_quotes_indexu_u_repfnsu_c0]: !a. h4/quote/_20_40ind__typequote3 (h4/quote/_20_40ind__typequote4 a) = a
% Assm [h4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/quote/_20_40ind__typequote0 x = h4/quote/_20_40ind__typequote3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (h4/ind__type/FCONS (h4/quote/_20_40ind__typequote4 x) _0)))
% Assm [h4s_primu_u_recs_wellfoundedu_u_def]: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. happ (happ R (happ f (h4/num/SUC n))) (happ f n))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_whiles_LEASTu_u_EXISTSu_u_IMP]: !p. (?n. happ p n) ==> happ p (h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (h4/while/LEAST p) ==> ~happ p n)
% Assm [h4s_arithmetics_FUNPOW0u_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (happ f x)
% Assm [h4s_whiles_OWHILEu_u_def]: !_0. (!G f s n. happ (happ (happ (happ _0 G) f) s) n <=> ~happ G (h4/arithmetic/FUNPOW f n s)) ==> (!s f G. ?v. (v <=> (?n. ~happ G (h4/arithmetic/FUNPOW f n s))) /\ h4/while/OWHILE G f s = h4/bool/COND v (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (happ (happ (happ _0 G) f) s)) s)) h4/option/NONE)
% Assm [h4s_arithmetics_FUNPOW0u_c0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_primu_u_recs_LESSu_u_MONO]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_PULLu_u_EXISTSu_c0]: !Q P. (?x. happ P x) ==> Q <=> (!x. happ P x ==> Q)
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_predu_u_sets_CARDu_u_BIGUNIONu_u_SAMEu_u_SIZEDu_u_SETS]: !s n. h4/pred__set/FINITE s /\ (!e. h4/bool/IN e s ==> h4/pred__set/FINITE e /\ h4/pred__set/CARD e = n) /\ (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s /\ ~(e1 = e2) ==> h4/pred__set/DISJOINT e1 e2) ==> h4/pred__set/CARD (h4/pred__set/BIGUNION s) = h4/arithmetic/_2A (h4/pred__set/CARD s) n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ]: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm [h4s_predu_u_sets_BIGUNIONu_u_INSERT]: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm [h4s_predu_u_sets_BIGUNIONu_u_EMPTY]: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_CARDu_u_EMPTY]: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_predu_u_sets_CARDu_u_INSERT]: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_DISJOINTu_u_DEF]: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_CARDu_u_UNIONu_u_EQN]: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/CARD (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t)) (h4/pred__set/CARD (h4/pred__set/INTER s t))
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Goal: !a1_27 a1 a0_27 a0. ~(happ (happ h4/canonical/SPplus a0) a1 = happ (happ h4/canonical/SPmult a0_27) a1_27)
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_Q1328742,TV_Q1328738]: ![V_f, V_g]: (![V_x]: s(TV_Q1328738,happ(s(t_fun(TV_Q1328742,TV_Q1328738),V_f),s(TV_Q1328742,V_x))) = s(TV_Q1328738,happ(s(t_fun(TV_Q1328742,TV_Q1328738),V_g),s(TV_Q1328742,V_x))) => s(t_fun(TV_Q1328742,TV_Q1328738),V_f) = s(t_fun(TV_Q1328742,TV_Q1328738),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_canonicals_SPmult0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult) = s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical8)).
fof(ah4s_canonicals_SPplus0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus) = s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical7)).
fof(ah4s_bools_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_canonicals_spolynomu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),V_a))))) = s(t_h4s_canonicals_spolynom(TV_u_27a),V_a)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical8u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical8),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_xi_))) = s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical7u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical7),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_xi_))) = s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_canonicals_spolynomu_u_distinctu_c1, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))).
fof(ah4s_canonicals_spolynomu_u_distinctu_c4, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))).
fof(ah4s_canonicals_spolynomu_u_distinctu_c3, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))).
fof(ah4s_canonicals_spolynomu_u_distinctu_c2, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))).
fof(ah4s_canonicals_spolynomu_u_sizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1)))))))).
fof(ah4s_canonicals_spolynomu_u_sizeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1)))))))).
fof(ah4s_canonicals_spolynomu_u_11u_c3, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0u_27))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1u_27))) <=> (s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0) = s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0u_27) & s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1) = s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1u_27)))).
fof(ah4s_canonicals_spolynomu_u_11u_c2, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0u_27))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1u_27))) <=> (s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0) = s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0u_27) & s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1) = s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1u_27)))).
fof(ah4s_canonicals_datatypeu_u_spolynom, axiom, ![TV_u_27a]: ![V_spolynom]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_bool),happ(s(t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_bool))),happ(s(t_fun(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_fun(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),t_bool)))),V_spolynom),s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar))),s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst))),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus))),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult))))))).
fof(ah4s_canonicals_SPconst0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst) = s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical6)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical6u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical6),s(TV_u_27a,V_x))) = s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
fof(ah4s_canonicals_SPvar0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar) = s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical5)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical5u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical5),s(t_h4s_quotes_index,V_x))) = s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_x),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
fof(ah4s_canonicals_spolynomu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27spolynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_canonicals_u_20u_40indu_u_typecanonical10(s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical9(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(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_n)))).
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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_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_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_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_canonicals_spolynomu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_27)))) <=> ![V_uu_27spolynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27spolynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_0),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_rep))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_canonicals_spolynomu_u_caseu_u_defu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_canonicals_spolynomu_u_case(s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f3))) = s(TV_u_27b,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f3),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1)))).
fof(ah4s_canonicals_spolynomu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_canonicals_spolynomu_u_case(s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f3))) = s(TV_u_27b,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f2),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1)))).
fof(ah4s_canonicals_spolynomu_u_distinctu_c0, axiom, ![TV_u_27a]: ![V_au_27, V_a]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_au_27))))).
fof(ah4s_canonicals_spolynomu_u_11u_c0, axiom, ![TV_u_27a]: ![V_au_27, V_a]: (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_au_27))) <=> s(t_h4s_quotes_index,V_a) = s(t_h4s_quotes_index,V_au_27))).
fof(ah4s_canonicals_spolynomu_u_sizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_quotes_indexu_u_size(s(t_h4s_quotes_index,V_a)))))).
fof(ah4s_canonicals_spolynomu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_canonicals_spolynomu_u_case(s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f3))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_a)))).
fof(ah4s_canonicals_spolynomu_u_11u_c1, axiom, ![TV_u_27a]: ![V_au_27, V_a]: (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_a))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_au_27))) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_au_27))).
fof(ah4s_canonicals_spolynomu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_canonicals_spolynomu_u_case(s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spvar),s(t_h4s_quotes_index,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27b)),V_f3))) = s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_h4s_quotes_index,V_a)))).
fof(ah4s_canonicals_spolynomu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a]: s(t_h4s_nums_num,h4s_canonicals_spolynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_canonicals_spolynom(TV_u_27a)),h4s_canonicals_spconst),s(TV_u_27a,V_a))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_distinctu_c2, axiom, ![TV_u_27a]: ![V_a2, V_a1u_27, V_a1, V_a0u_27, V_a0]: ~ (s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_canonicals_consu_u_monom),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_a1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a2))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_canonicals_consu_u_varlist),s(t_h4s_lists_list(t_h4s_quotes_index),V_a0u_27))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a1u_27))))).
fof(ah4s_sums_sumu_u_distinct, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: ~ (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical2),s(t_h4s_lists_list(t_h4s_quotes_index),V_x))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x0))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27canonicalu_u_sumu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))) | (?[V_a0, V_a1, V_a2]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_lists_list(t_h4s_quotes_index),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a2))))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),V_a0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a1))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r)))).
fof(ah4s_canonicals_Consu_u_monom0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_canonicals_consu_u_monom) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_canonicals_u_20u_40indu_u_typecanonical1)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical0) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))).
fof(ah4s_canonicals_Nilu_u_monom0, axiom, ![TV_u_27a]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical0)).
fof(ah4s_canonicals_Consu_u_varlist0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_canonicals_consu_u_varlist) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_canonicals_u_20u_40indu_u_typecanonical2)).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_x, V_x0, V_x1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_canonicals_u_20u_40indu_u_typecanonical1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),V_x0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x1))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))))).
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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_combins_UPD11u_u_SAMEu_u_BASE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) <=> ((s(TV_u_27a,V_a) = s(TV_u_27a,V_b) & s(TV_u_27b,V_c) = s(TV_u_27b,V_d)) | (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b)) & (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f) & s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)))))).
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_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
fof(ah4s_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_combins_UPDATEu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_b, V_a, V_x, V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x0)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))))))).
fof(ah4s_bools_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_ODD0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_quotes_indexu_u_distinctu_c1, axiom, ![V_a]: ~ (s(t_h4s_quotes_index,happ(s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_leftu_u_idx),s(t_h4s_quotes_index,V_a))) = s(t_h4s_quotes_index,h4s_quotes_endu_u_idx))).
fof(ah4s_whiles_OWHILEu_u_IND, axiom, ![TV_u_27a]: ![V_f, V_P, V_G]: ((![V_s]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),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_P),s(TV_u_27a,V_s))),s(TV_u_27a,V_s))))) & ![V_s1, V_s2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,V_s1)))) & 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_P),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s1))))),s(TV_u_27a,V_s2))))) => 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_P),s(TV_u_27a,V_s1))),s(TV_u_27a,V_s2)))))) => ![V_s1, V_s2]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s1))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_s2))) => 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_P),s(TV_u_27a,V_s1))),s(TV_u_27a,V_s2))))))).
fof(ah4s_quotes_Endu_u_idx0, axiom, s(t_h4s_quotes_index,h4s_quotes_endu_u_idx) = s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote2)).
fof(ah4s_quotes_Rightu_u_idx0, axiom, s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_rightu_u_idx) = s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_u_20u_40indu_u_typequote1)).
fof(ah4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote2u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote2) = s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))).
fof(ah4s_quotes_Leftu_u_idx0, axiom, s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_leftu_u_idx) = s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_u_20u_40indu_u_typequote0)).
fof(ah4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote1u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_quotes_index,happ(s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_u_20u_40indu_u_typequote1),s(t_h4s_quotes_index,V_x))) = s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),h4s_quotes_u_20u_40indu_u_typequote4(s(t_h4s_quotes_index,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))))).
fof(ah4s_quotes_indexu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27indexu_27]: (![V_a00]: ((?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27indexu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27indexu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))) | s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27indexu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27indexu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_bool),h4s_quotes_u_20u_40indu_u_typequote4(s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote3(s(t_h4s_indu_u_types_recspace(t_bool),V_r))))) = s(t_h4s_indu_u_types_recspace(t_bool),V_r)))).
fof(ah4s_quotes_indexu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_quotes_u_20u_40indu_u_typequote4(s(t_h4s_quotes_index,V_a))))) = s(t_h4s_quotes_index,V_a)).
fof(ah4s_quotes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typequote0u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_quotes_index,happ(s(t_fun(t_h4s_quotes_index,t_h4s_quotes_index),h4s_quotes_u_20u_40indu_u_typequote0),s(t_h4s_quotes_index,V_x))) = s(t_h4s_quotes_index,h4s_quotes_u_20u_40indu_u_typequote3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),h4s_quotes_u_20u_40indu_u_typequote4(s(t_h4s_quotes_index,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))))).
fof(ah4s_primu_u_recs_wellfoundedu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_primu_u_recs_wellfounded(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ~ (?[V_f]: ![V_n]: 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(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(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_n))))))))).
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_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_whiles_LEASTu_u_EXISTSu_u_IMP, axiom, ![V_p]: (?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_n)))) => (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) & ![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_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) => ~ (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_FUNPOW0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(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(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_whiles_OWHILEu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_G, V_f, V_s, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))),s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) => ![V_s, V_f, V_G]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) & s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))).
fof(ah4s_arithmetics_FUNPOW0u_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_primu_u_recs_LESSu_u_MONO, 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_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)))))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_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_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_bools_PULLu_u_EXISTSu_c0, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_bools_DISJu_u_COMM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_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_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_predu_u_sets_CARDu_u_BIGUNIONu_u_SAMEu_u_SIZEDu_u_SETS, axiom, ![TV_u_27a]: ![V_s, V_n]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & (![V_e]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) => (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_e)))) & s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_e))) = s(t_h4s_nums_num,V_n))) & ![V_e1, V_e2]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_e1) = s(t_fun(TV_u_27a,t_bool),V_e2)))) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_e1),s(t_fun(TV_u_27a,t_bool),V_e2))))))) => s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s))),s(t_h4s_nums_num,V_n))))).
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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ, axiom, ![TV_u_27a]: ![V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_BIGUNIONu_u_INSERT, axiom, ![TV_u_27a]: ![V_s, V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_insert(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))).
fof(ah4s_predu_u_sets_BIGUNIONu_u_EMPTY, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_CARDu_u_EMPTY, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_predu_u_sets_CARDu_u_INSERT, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_DISJOINTu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_CARDu_u_UNIONu_u_EQN, axiom, ![TV_u_27a]: ![V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))) => s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ch4s_canonicals_spolynomu_u_distinctu_c5, conjecture, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: ~ (s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spplus),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1))) = s(t_h4s_canonicals_spolynom(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_spolynom(TV_u_27a))),h4s_canonicals_spmult),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a0u_27))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_a1u_27))))).
