TPTP Problem File: ANA143_1.008.128.p
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%------------------------------------------------------------------------------
% File : ANA143_1.008.128 : TPTP v8.2.0. Released v8.2.0.
% Domain : Number theory
% Problem : linear_limit_n8_m128__1
% Version : Especial.
% English : lim(f_1 x) = l_1
% ...
% lim(f_n x) = l_n
% =================
% lim(k_1 f_{i_1}(x) + ... + k_m f_{i_m}(x)) = k1 l_{i_1} ... k_m l_{i_m}
%
% with n = 8
% m = 128
% k_i in Q
% i_j in {1..n}
% Refs : [Sch22] Schoisswohl (2022), Email to G. Sutcliffe
% : [KK+23] Korovin et al. (2023), ALASCA: Reasoning in Quantified
% Source : [Sch22]
% Names : linear_limit_n8_m128__1.smt2 [Sch22]
% Status : Theorem
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 26 ( 0 unt; 17 typ; 0 def)
% Number of atoms : 99 ( 9 equ)
% Maximal formula atoms : 11 ( 11 avg)
% Number of connectives : 117 ( 27 ~; 0 |; 36 &)
% ( 0 <=>; 54 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 11 avg)
% Maximal term depth : 133 ( 11 avg)
% Number arithmetic : 5917 ( 90 atm;3714 fun;2086 num; 27 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 115 ( 17 usr; 102 con; 0-2 aty)
% Number of variables : 27 ( 18 !; 9 ?; 27 :)
% SPC : TF0_THM_EQU_ARI
% Comments : Translated from SMT UFLRA by SMTtoTPTP.
%------------------------------------------------------------------------------
%% Declarations:
tff(l4,type,
l4: $real ).
tff(f5,type,
f5: $real > $real ).
tff(l5,type,
l5: $real ).
tff(f6,type,
f6: $real > $real ).
tff(f0,type,
f0: $real > $real ).
tff(l0,type,
l0: $real ).
tff(f4,type,
f4: $real > $real ).
tff(f2,type,
f2: $real > $real ).
tff(l2,type,
l2: $real ).
tff(a,type,
a: $real ).
tff(f7,type,
f7: $real > $real ).
tff(f1,type,
f1: $real > $real ).
tff(l1,type,
l1: $real ).
tff(l7,type,
l7: $real ).
tff(l3,type,
l3: $real ).
tff(f3,type,
f3: $real > $real ).
tff(l6,type,
l6: $real ).
%% Assertions:
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f0(x) - l0) ≥ 0.0) (f0(x) - l0) else -(f0(x) - l0)) < epsilon))))
tff(formula_1,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l0),0.0)
=> $less($difference(f0(X),l0),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l0),0.0)
=> $less($uminus($difference(f0(X),l0)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f1(x) - l1) ≥ 0.0) (f1(x) - l1) else -(f1(x) - l1)) < epsilon))))
tff(formula_2,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l1),0.0)
=> $less($difference(f1(X),l1),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l1),0.0)
=> $less($uminus($difference(f1(X),l1)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f2(x) - l2) ≥ 0.0) (f2(x) - l2) else -(f2(x) - l2)) < epsilon))))
tff(formula_3,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f2(X),l2),0.0)
=> $less($difference(f2(X),l2),Epsilon) )
& ( ~ $greatereq($difference(f2(X),l2),0.0)
=> $less($uminus($difference(f2(X),l2)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f3(x) - l3) ≥ 0.0) (f3(x) - l3) else -(f3(x) - l3)) < epsilon))))
tff(formula_4,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f3(X),l3),0.0)
=> $less($difference(f3(X),l3),Epsilon) )
& ( ~ $greatereq($difference(f3(X),l3),0.0)
=> $less($uminus($difference(f3(X),l3)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f4(x) - l4) ≥ 0.0) (f4(x) - l4) else -(f4(x) - l4)) < epsilon))))
tff(formula_5,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f4(X),l4),0.0)
=> $less($difference(f4(X),l4),Epsilon) )
& ( ~ $greatereq($difference(f4(X),l4),0.0)
=> $less($uminus($difference(f4(X),l4)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f5(x) - l5) ≥ 0.0) (f5(x) - l5) else -(f5(x) - l5)) < epsilon))))
tff(formula_6,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f5(X),l5),0.0)
=> $less($difference(f5(X),l5),Epsilon) )
& ( ~ $greatereq($difference(f5(X),l5),0.0)
=> $less($uminus($difference(f5(X),l5)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f6(x) - l6) ≥ 0.0) (f6(x) - l6) else -(f6(x) - l6)) < epsilon))))
tff(formula_7,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f6(X),l6),0.0)
=> $less($difference(f6(X),l6),Epsilon) )
& ( ~ $greatereq($difference(f6(X),l6),0.0)
=> $less($uminus($difference(f6(X),l6)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if ((f7(x) - l7) ≥ 0.0) (f7(x) - l7) else -(f7(x) - l7)) < epsilon))))
tff(formula_8,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( $greatereq($difference(f7(X),l7),0.0)
=> $less($difference(f7(X),l7),Epsilon) )
& ( ~ $greatereq($difference(f7(X),l7),0.0)
=> $less($uminus($difference(f7(X),l7)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = a) ∧ ((if ((x - a) ≥ 0.0) (x - a) else -(x - a)) < delta)) ⇒ ((if (((((52.0 / 80.0) * f6(x)) + (-(20.0 / 53.0) * f0(x)) + ((14.0 / 81.0) * f2(x)) + ((61.0 / 5.0) * f2(x)) + (-(96.0 / 63.0) * f0(x)) + (-(19.0 / 27.0) * f2(x)) + (-(30.0 / 40.0) * f6(x)) + (-(12.0 / 84.0) * f7(x)) + (-(83.0 / 100.0) * f1(x)) + ((67.0 / 99.0) * f0(x)) + ((83.0 / 23.0) * f5(x)) + (-(45.0 / 32.0) * f5(x)) + (-(83.0 / 100.0) * f5(x)) + ((43.0 / 26.0) * f5(x)) + ((33.0 / 30.0) * f6(x)) + (-(67.0 / 74.0) * f6(x)) + ((95.0 / 60.0) * f6(x)) + (-(79.0 / 46.0) * f1(x)) + ((76.0 / 75.0) * f3(x)) + (-(82.0 / 18.0) * f7(x)) + (-(14.0 / 6.0) * f0(x)) + (-(100.0 / 57.0) * f2(x)) + ((89.0 / 58.0) * f0(x)) + (-(32.0 / 28.0) * f6(x)) + (-(23.0 / 38.0) * f3(x)) + ((61.0 / 76.0) * f3(x)) + (-(97.0 / 25.0) * f7(x)) + (-(52.0 / 85.0) * f7(x)) + ((61.0 / 58.0) * f4(x)) + (-(41.0 / 99.0) * f0(x)) + (-(100.0 / 8.0) * f3(x)) + (-(71.0 / 13.0) * f3(x)) + (18.0 * f6(x)) + (-(25.0 / 40.0) * f7(x)) + (-(1.0 / 53.0) * f4(x)) + (-(84.0 / 37.0) * f4(x)) + (-(57.0 / 87.0) * f0(x)) + (-(9.0 / 98.0) * f4(x)) + ((34.0 / 9.0) * f3(x)) + ((30.0 / 78.0) * f7(x)) + ((84.0 / 55.0) * f2(x)) + (-(23.0 / 95.0) * f1(x)) + ((63.0 / 84.0) * f6(x)) + (-(6.0 / 37.0) * f0(x)) + (-(52.0 / 88.0) * f7(x)) + (-(33.0 / 14.0) * f5(x)) + ((7.0 / 39.0) * f2(x)) + (-(18.0 / 96.0) * f3(x)) + (-(0.0 / 20.0) * f4(x)) + (-(13.0 / 42.0) * f2(x)) + (-(12.0 / 5.0) * f2(x)) + ((32.0 / 24.0) * f1(x)) + ((40.0 / 79.0) * f7(x)) + ((12.0 / 36.0) * f4(x)) + (-(69.0 / 45.0) * f7(x)) + (-(5.0 / 93.0) * f5(x)) + (-(3.0 / 73.0) * f0(x)) + ((92.0 / 83.0) * f2(x)) + (-(4.0 / 43.0) * f2(x)) + (-(1.0 / 70.0) * f5(x)) + (-(58.0 / 87.0) * f0(x)) + ((47.0 / 7.0) * f1(x)) + (-(59.0 / 66.0) * f6(x)) + (-(30.0 / 57.0) * f0(x)) + (-(10.0 / 78.0) * f0(x)) + ((79.0 / 77.0) * f3(x)) + ((29.0 / 81.0) * f4(x)) + (-(69.0 / 72.0) * f3(x)) + ((56.0 / 5.0) * f3(x)) + (-(43.0 / 93.0) * f1(x)) + (-(2.0 / 46.0) * f2(x)) + ((74.0 / 51.0) * f6(x)) + (-(85.0 / 87.0) * f4(x)) + (-(77.0 / 12.0) * f0(x)) + ((86.0 / 69.0) * f5(x)) + (-(45.0 / 44.0) * f1(x)) + (-(15.0 / 33.0) * f7(x)) + (-(63.0 / 45.0) * f3(x)) + (-(41.0 / 92.0) * f6(x)) + ((91.0 / 91.0) * f7(x)) + (-(17.0 / 12.0) * f0(x)) + (-(83.0 / 36.0) * f0(x)) + ((77.0 / 46.0) * f6(x)) + ((49.0 / 86.0) * f3(x)) + (-(82.0 / 88.0) * f2(x)) + (-(33.0 / 88.0) * f7(x)) + (-(24.0 / 67.0) * f3(x)) + (-(71.0 / 97.0) * f5(x)) + ((15.0 / 13.0) * f5(x)) + ((47.0 / 43.0) * f4(x)) + (-(61.0 / 99.0) * f0(x)) + ((63.0 / 69.0) * f5(x)) + ((72.0 / 38.0) * f4(x)) + (-(40.0 / 38.0) * f4(x)) + ((4.0 / 77.0) * f7(x)) + (-(34.0 / 97.0) * f4(x)) + ((43.0 / 97.0) * f4(x)) + (-(60.0 / 19.0) * f1(x)) + ((15.0 / 3.0) * f1(x)) + ((17.0 / 66.0) * f6(x)) + ((59.0 / 57.0) * f0(x)) + (54.0 * f5(x)) + ((8.0 / 2.0) * f5(x)) + ((59.0 / 85.0) * f0(x)) + (-(57.0 / 38.0) * f7(x)) + ((42.0 / 3.0) * f7(x)) + ((52.0 / 48.0) * f0(x)) + (-(86.0 / 10.0) * f7(x)) + ((78.0 / 63.0) * f1(x)) + (-(11.0 / 59.0) * f2(x)) + ((57.0 / 29.0) * f3(x)) + (-(34.0 / 35.0) * f1(x)) + ((18.0 / 84.0) * f2(x)) + (-(68.0 / 22.0) * f4(x)) + (-(95.0 / 56.0) * f6(x)) + ((34.0 / 43.0) * f7(x)) + ((73.0 / 85.0) * f2(x)) + ((98.0 / 69.0) * f7(x)) + ((3.0 / 6.0) * f7(x)) + ((84.0 / 92.0) * f5(x)) + (-(41.0 / 81.0) * f4(x)) + (-(98.0 / 50.0) * f1(x)) + (-(76.0 / 30.0) * f0(x)) + (-(47.0 / 32.0) * f2(x)) + ((59.0 / 6.0) * f3(x)) + ((26.0 / 14.0) * f2(x)) + (-(58.0 / 40.0) * f0(x)) + ((61.0 / 28.0) * f7(x))) - (((52.0 / 80.0) * l6) + (-(20.0 / 53.0) * l0) + ((14.0 / 81.0) * l2) + ((61.0 / 5.0) * l2) + (-(96.0 / 63.0) * l0) + (-(19.0 / 27.0) * l2) + (-(30.0 / 40.0) * l6) + (-(12.0 / 84.0) * l7) + (-(83.0 / 100.0) * l1) + ((67.0 / 99.0) * l0) + ((83.0 / 23.0) * l5) + (-(45.0 / 32.0) * l5) + (-(83.0 / 100.0) * l5) + ((43.0 / 26.0) * l5) + ((33.0 / 30.0) * l6) + (-(67.0 / 74.0) * l6) + ((95.0 / 60.0) * l6) + (-(79.0 / 46.0) * l1) + ((76.0 / 75.0) * l3) + (-(82.0 / 18.0) * l7) + (-(14.0 / 6.0) * l0) + (-(100.0 / 57.0) * l2) + ((89.0 / 58.0) * l0) + (-(32.0 / 28.0) * l6) + (-(23.0 / 38.0) * l3) + ((61.0 / 76.0) * l3) + (-(97.0 / 25.0) * l7) + (-(52.0 / 85.0) * l7) + ((61.0 / 58.0) * l4) + (-(41.0 / 99.0) * l0) + (-(100.0 / 8.0) * l3) + (-(71.0 / 13.0) * l3) + (18.0 * l6) + (-(25.0 / 40.0) * l7) + (-(1.0 / 53.0) * l4) + (-(84.0 / 37.0) * l4) + (-(57.0 / 87.0) * l0) + (-(9.0 / 98.0) * l4) + ((34.0 / 9.0) * l3) + ((30.0 / 78.0) * l7) + ((84.0 / 55.0) * l2) + (-(23.0 / 95.0) * l1) + ((63.0 / 84.0) * l6) + (-(6.0 / 37.0) * l0) + (-(52.0 / 88.0) * l7) + (-(33.0 / 14.0) * l5) + ((7.0 / 39.0) * l2) + (-(18.0 / 96.0) * l3) + (-(0.0 / 20.0) * l4) + (-(13.0 / 42.0) * l2) + (-(12.0 / 5.0) * l2) + ((32.0 / 24.0) * l1) + ((40.0 / 79.0) * l7) + ((12.0 / 36.0) * l4) + (-(69.0 / 45.0) * l7) + (-(5.0 / 93.0) * l5) + (-(3.0 / 73.0) * l0) + ((92.0 / 83.0) * l2) + (-(4.0 / 43.0) * l2) + (-(1.0 / 70.0) * l5) + (-(58.0 / 87.0) * l0) + ((47.0 / 7.0) * l1) + (-(59.0 / 66.0) * l6) + (-(30.0 / 57.0) * l0) + (-(10.0 / 78.0) * l0) + ((79.0 / 77.0) * l3) + ((29.0 / 81.0) * l4) + (-(69.0 / 72.0) * l3) + ((56.0 / 5.0) * l3) + (-(43.0 / 93.0) * l1) + (-(2.0 / 46.0) * l2) + ((74.0 / 51.0) * l6) + (-(85.0 / 87.0) * l4) + (-(77.0 / 12.0) * l0) + ((86.0 / 69.0) * l5) + (-(45.0 / 44.0) * l1) + (-(15.0 / 33.0) * l7) + (-(63.0 / 45.0) * l3) + (-(41.0 / 92.0) * l6) + ((91.0 / 91.0) * l7) + (-(17.0 / 12.0) * l0) + (-(83.0 / 36.0) * l0) + ((77.0 / 46.0) * l6) + ((49.0 / 86.0) * l3) + (-(82.0 / 88.0) * l2) + (-(33.0 / 88.0) * l7) + (-(24.0 / 67.0) * l3) + (-(71.0 / 97.0) * l5) + ((15.0 / 13.0) * l5) + ((47.0 / 43.0) * l4) + (-(61.0 / 99.0) * l0) + ((63.0 / 69.0) * l5) + ((72.0 / 38.0) * l4) + (-(40.0 / 38.0) * l4) + ((4.0 / 77.0) * l7) + (-(34.0 / 97.0) * l4) + ((43.0 / 97.0) * l4) + (-(60.0 / 19.0) * l1) + ((15.0 / 3.0) * l1) + ((17.0 / 66.0) * l6) + ((59.0 / 57.0) * l0) + (54.0 * l5) + ((8.0 / 2.0) * l5) + ((59.0 / 85.0) * l0) + (-(57.0 / 38.0) * l7) + ((42.0 / 3.0) * l7) + ((52.0 / 48.0) * l0) + (-(86.0 / 10.0) * l7) + ((78.0 / 63.0) * l1) + (-(11.0 / 59.0) * l2) + ((57.0 / 29.0) * l3) + (-(34.0 / 35.0) * l1) + ((18.0 / 84.0) * l2) + (-(68.0 / 22.0) * l4) + (-(95.0 / 56.0) * l6) + ((34.0 / 43.0) * l7) + ((73.0 / 85.0) * l2) + ((98.0 / 69.0) * l7) + ((3.0 / 6.0) * l7) + ((84.0 / 92.0) * l5) + (-(41.0 / 81.0) * l4) + (-(98.0 / 50.0) * l1) + (-(76.0 / 30.0) * l0) + (-(47.0 / 32.0) * l2) + ((59.0 / 6.0) * l3) + ((26.0 / 14.0) * l2) + (-(58.0 / 40.0) * l0) + ((61.0 / 28.0) * l7))) ≥ 0.0) ((((52.0 / 80.0) * f6(x)) + (-(20.0 / 53.0) * f0(x)) + ((14.0 / 81.0) * f2(x)) + ((61.0 / 5.0) * f2(x)) + (-(96.0 / 63.0) * f0(x)) + (-(19.0 / 27.0) * f2(x)) + (-(30.0 / 40.0) * f6(x)) + (-(12.0 / 84.0) * f7(x)) + (-(83.0 / 100.0) * f1(x)) + ((67.0 / 99.0) * f0(x)) + ((83.0 / 23.0) * f5(x)) + (-(45.0 / 32.0) * f5(x)) + (-(83.0 / 100.0) * f5(x)) + ((43.0 / 26.0) * f5(x)) + ((33.0 / 30.0) * f6(x)) + (-(67.0 / 74.0) * f6(x)) + ((95.0 / 60.0) * f6(x)) + (-(79.0 / 46.0) * f1(x)) + ((76.0 / 75.0) * f3(x)) + (-(82.0 / 18.0) * f7(x)) + (-(14.0 / 6.0) * f0(x)) + (-(100.0 / 57.0) * f2(x)) + ((89.0 / 58.0) * f0(x)) + (-(32.0 / 28.0) * f6(x)) + (-(23.0 / 38.0) * f3(x)) + ((61.0 / 76.0) * f3(x)) + (-(97.0 / 25.0) * f7(x)) + (-(52.0 / 85.0) * f7(x)) + ((61.0 / 58.0) * f4(x)) + (-(41.0 / 99.0) * f0(x)) + (-(100.0 / 8.0) * f3(x)) + (-(71.0 / 13.0) * f3(x)) + (18.0 * f6(x)) + (-(25.0 / 40.0) * f7(x)) + (-(1.0 / 53.0) * f4(x)) + (-(84.0 / 37.0) * f4(x)) + (-(57.0 / 87.0) * f0(x)) + (-(9.0 / 98.0) * f4(x)) + ((34.0 / 9.0) * f3(x)) + ((30.0 / 78.0) * f7(x)) + ((84.0 / 55.0) * f2(x)) + (-(23.0 / 95.0) * f1(x)) + ((63.0 / 84.0) * f6(x)) + (-(6.0 / 37.0) * f0(x)) + (-(52.0 / 88.0) * f7(x)) + (-(33.0 / 14.0) * f5(x)) + ((7.0 / 39.0) * f2(x)) + (-(18.0 / 96.0) * f3(x)) + (-(0.0 / 20.0) * f4(x)) + (-(13.0 / 42.0) * f2(x)) + (-(12.0 / 5.0) * f2(x)) + ((32.0 / 24.0) * f1(x)) + ((40.0 / 79.0) * f7(x)) + ((12.0 / 36.0) * f4(x)) + (-(69.0 / 45.0) * f7(x)) + (-(5.0 / 93.0) * f5(x)) + (-(3.0 / 73.0) * f0(x)) + ((92.0 / 83.0) * f2(x)) + (-(4.0 / 43.0) * f2(x)) + (-(1.0 / 70.0) * f5(x)) + (-(58.0 / 87.0) * f0(x)) + ((47.0 / 7.0) * f1(x)) + (-(59.0 / 66.0) * f6(x)) + (-(30.0 / 57.0) * f0(x)) + (-(10.0 / 78.0) * f0(x)) + ((79.0 / 77.0) * f3(x)) + ((29.0 / 81.0) * f4(x)) + (-(69.0 / 72.0) * f3(x)) + ((56.0 / 5.0) * f3(x)) + (-(43.0 / 93.0) * f1(x)) + (-(2.0 / 46.0) * f2(x)) + ((74.0 / 51.0) * f6(x)) + (-(85.0 / 87.0) * f4(x)) + (-(77.0 / 12.0) * f0(x)) + ((86.0 / 69.0) * f5(x)) + (-(45.0 / 44.0) * f1(x)) + (-(15.0 / 33.0) * f7(x)) + (-(63.0 / 45.0) * f3(x)) + (-(41.0 / 92.0) * f6(x)) + ((91.0 / 91.0) * f7(x)) + (-(17.0 / 12.0) * f0(x)) + (-(83.0 / 36.0) * f0(x)) + ((77.0 / 46.0) * f6(x)) + ((49.0 / 86.0) * f3(x)) + (-(82.0 / 88.0) * f2(x)) + (-(33.0 / 88.0) * f7(x)) + (-(24.0 / 67.0) * f3(x)) + (-(71.0 / 97.0) * f5(x)) + ((15.0 / 13.0) * f5(x)) + ((47.0 / 43.0) * f4(x)) + (-(61.0 / 99.0) * f0(x)) + ((63.0 / 69.0) * f5(x)) + ((72.0 / 38.0) * f4(x)) + (-(40.0 / 38.0) * f4(x)) + ((4.0 / 77.0) * f7(x)) + (-(34.0 / 97.0) * f4(x)) + ((43.0 / 97.0) * f4(x)) + (-(60.0 / 19.0) * f1(x)) + ((15.0 / 3.0) * f1(x)) + ((17.0 / 66.0) * f6(x)) + ((59.0 / 57.0) * f0(x)) + (54.0 * f5(x)) + ((8.0 / 2.0) * f5(x)) + ((59.0 / 85.0) * f0(x)) + (-(57.0 / 38.0) * f7(x)) + ((42.0 / 3.0) * f7(x)) + ((52.0 / 48.0) * f0(x)) + (-(86.0 / 10.0) * f7(x)) + ((78.0 / 63.0) * f1(x)) + (-(11.0 / 59.0) * f2(x)) + ((57.0 / 29.0) * f3(x)) + (-(34.0 / 35.0) * f1(x)) + ((18.0 / 84.0) * f2(x)) + (-(68.0 / 22.0) * f4(x)) + (-(95.0 / 56.0) * f6(x)) + ((34.0 / 43.0) * f7(x)) + ((73.0 / 85.0) * f2(x)) + ((98.0 / 69.0) * f7(x)) + ((3.0 / 6.0) * f7(x)) + ((84.0 / 92.0) * f5(x)) + (-(41.0 / 81.0) * f4(x)) + (-(98.0 / 50.0) * f1(x)) + (-(76.0 / 30.0) * f0(x)) + (-(47.0 / 32.0) * f2(x)) + ((59.0 / 6.0) * f3(x)) + ((26.0 / 14.0) * f2(x)) + (-(58.0 / 40.0) * f0(x)) + ((61.0 / 28.0) * f7(x))) - (((52.0 / 80.0) * l6) + (-(20.0 / 53.0) * l0) + ((14.0 / 81.0) * l2) + ((61.0 / 5.0) * l2) + (-(96.0 / 63.0) * l0) + (-(19.0 / 27.0) * l2) + (-(30.0 / 40.0) * l6) + (-(12.0 / 84.0) * l7) + (-(83.0 / 100.0) * l1) + ((67.0 / 99.0) * l0) + ((83.0 / 23.0) * l5) + (-(45.0 / 32.0) * l5) + (-(83.0 / 100.0) * l5) + ((43.0 / 26.0) * l5) + ((33.0 / 30.0) * l6) + (-(67.0 / 74.0) * l6) + ((95.0 / 60.0) * l6) + (-(79.0 / 46.0) * l1) + ((76.0 / 75.0) * l3) + (-(82.0 / 18.0) * l7) + (-(14.0 / 6.0) * l0) + (-(100.0 / 57.0) * l2) + ((89.0 / 58.0) * l0) + (-(32.0 / 28.0) * l6) + (-(23.0 / 38.0) * l3) + ((61.0 / 76.0) * l3) + (-(97.0 / 25.0) * l7) + (-(52.0 / 85.0) * l7) + ((61.0 / 58.0) * l4) + (-(41.0 / 99.0) * l0) + (-(100.0 / 8.0) * l3) + (-(71.0 / 13.0) * l3) + (18.0 * l6) + (-(25.0 / 40.0) * l7) + (-(1.0 / 53.0) * l4) + (-(84.0 / 37.0) * l4) + (-(57.0 / 87.0) * l0) + (-(9.0 / 98.0) * l4) + ((34.0 / 9.0) * l3) + ((30.0 / 78.0) * l7) + ((84.0 / 55.0) * l2) + (-(23.0 / 95.0) * l1) + ((63.0 / 84.0) * l6) + (-(6.0 / 37.0) * l0) + (-(52.0 / 88.0) * l7) + (-(33.0 / 14.0) * l5) + ((7.0 / 39.0) * l2) + (-(18.0 / 96.0) * l3) + (-(0.0 / 20.0) * l4) + (-(13.0 / 42.0) * l2) + (-(12.0 / 5.0) * l2) + ((32.0 / 24.0) * l1) + ((40.0 / 79.0) * l7) + ((12.0 / 36.0) * l4) + (-(69.0 / 45.0) * l7) + (-(5.0 / 93.0) * l5) + (-(3.0 / 73.0) * l0) + ((92.0 / 83.0) * l2) + (-(4.0 / 43.0) * l2) + (-(1.0 / 70.0) * l5) + (-(58.0 / 87.0) * l0) + ((47.0 / 7.0) * l1) + (-(59.0 / 66.0) * l6) + (-(30.0 / 57.0) * l0) + (-(10.0 / 78.0) * l0) + ((79.0 / 77.0) * l3) + ((29.0 / 81.0) * l4) + (-(69.0 / 72.0) * l3) + ((56.0 / 5.0) * l3) + (-(43.0 / 93.0) * l1) + (-(2.0 / 46.0) * l2) + ((74.0 / 51.0) * l6) + (-(85.0 / 87.0) * l4) + (-(77.0 / 12.0) * l0) + ((86.0 / 69.0) * l5) + (-(45.0 / 44.0) * l1) + (-(15.0 / 33.0) * l7) + (-(63.0 / 45.0) * l3) + (-(41.0 / 92.0) * l6) + ((91.0 / 91.0) * l7) + (-(17.0 / 12.0) * l0) + (-(83.0 / 36.0) * l0) + ((77.0 / 46.0) * l6) + ((49.0 / 86.0) * l3) + (-(82.0 / 88.0) * l2) + (-(33.0 / 88.0) * l7) + (-(24.0 / 67.0) * l3) + (-(71.0 / 97.0) * l5) + ((15.0 / 13.0) * l5) + ((47.0 / 43.0) * l4) + (-(61.0 / 99.0) * l0) + ((63.0 / 69.0) * l5) + ((72.0 / 38.0) * l4) + (-(40.0 / 38.0) * l4) + ((4.0 / 77.0) * l7) + (-(34.0 / 97.0) * l4) + ((43.0 / 97.0) * l4) + (-(60.0 / 19.0) * l1) + ((15.0 / 3.0) * l1) + ((17.0 / 66.0) * l6) + ((59.0 / 57.0) * l0) + (54.0 * l5) + ((8.0 / 2.0) * l5) + ((59.0 / 85.0) * l0) + (-(57.0 / 38.0) * l7) + ((42.0 / 3.0) * l7) + ((52.0 / 48.0) * l0) + (-(86.0 / 10.0) * l7) + ((78.0 / 63.0) * l1) + (-(11.0 / 59.0) * l2) + ((57.0 / 29.0) * l3) + (-(34.0 / 35.0) * l1) + ((18.0 / 84.0) * l2) + (-(68.0 / 22.0) * l4) + (-(95.0 / 56.0) * l6) + ((34.0 / 43.0) * l7) + ((73.0 / 85.0) * l2) + ((98.0 / 69.0) * l7) + ((3.0 / 6.0) * l7) + ((84.0 / 92.0) * l5) + (-(41.0 / 81.0) * l4) + (-(98.0 / 50.0) * l1) + (-(76.0 / 30.0) * l0) + (-(47.0 / 32.0) * l2) + ((59.0 / 6.0) * l3) + ((26.0 / 14.0) * l2) + (-(58.0 / 40.0) * l0) + ((61.0 / 28.0) * l7))) else -((((52.0 / 80.0) * f6(x)) + (-(20.0 / 53.0) * f0(x)) + ((14.0 / 81.0) * f2(x)) + ((61.0 / 5.0) * f2(x)) + (-(96.0 / 63.0) * f0(x)) + (-(19.0 / 27.0) * f2(x)) + (-(30.0 / 40.0) * f6(x)) + (-(12.0 / 84.0) * f7(x)) + (-(83.0 / 100.0) * f1(x)) + ((67.0 / 99.0) * f0(x)) + ((83.0 / 23.0) * f5(x)) + (-(45.0 / 32.0) * f5(x)) + (-(83.0 / 100.0) * f5(x)) + ((43.0 / 26.0) * f5(x)) + ((33.0 / 30.0) * f6(x)) + (-(67.0 / 74.0) * f6(x)) + ((95.0 / 60.0) * f6(x)) + (-(79.0 / 46.0) * f1(x)) + ((76.0 / 75.0) * f3(x)) + (-(82.0 / 18.0) * f7(x)) + (-(14.0 / 6.0) * f0(x)) + (-(100.0 / 57.0) * f2(x)) + ((89.0 / 58.0) * f0(x)) + (-(32.0 / 28.0) * f6(x)) + (-(23.0 / 38.0) * f3(x)) + ((61.0 / 76.0) * f3(x)) + (-(97.0 / 25.0) * f7(x)) + (-(52.0 / 85.0) * f7(x)) + ((61.0 / 58.0) * f4(x)) + (-(41.0 / 99.0) * f0(x)) + (-(100.0 / 8.0) * f3(x)) + (-(71.0 / 13.0) * f3(x)) + (18.0 * f6(x)) + (-(25.0 / 40.0) * f7(x)) + (-(1.0 / 53.0) * f4(x)) + (-(84.0 / 37.0) * f4(x)) + (-(57.0 / 87.0) * f0(x)) + (-(9.0 / 98.0) * f4(x)) + ((34.0 / 9.0) * f3(x)) + ((30.0 / 78.0) * f7(x)) + ((84.0 / 55.0) * f2(x)) + (-(23.0 / 95.0) * f1(x)) + ((63.0 / 84.0) * f6(x)) + (-(6.0 / 37.0) * f0(x)) + (-(52.0 / 88.0) * f7(x)) + (-(33.0 / 14.0) * f5(x)) + ((7.0 / 39.0) * f2(x)) + (-(18.0 / 96.0) * f3(x)) + (-(0.0 / 20.0) * f4(x)) + (-(13.0 / 42.0) * f2(x)) + (-(12.0 / 5.0) * f2(x)) + ((32.0 / 24.0) * f1(x)) + ((40.0 / 79.0) * f7(x)) + ((12.0 / 36.0) * f4(x)) + (-(69.0 / 45.0) * f7(x)) + (-(5.0 / 93.0) * f5(x)) + (-(3.0 / 73.0) * f0(x)) + ((92.0 / 83.0) * f2(x)) + (-(4.0 / 43.0) * f2(x)) + (-(1.0 / 70.0) * f5(x)) + (-(58.0 / 87.0) * f0(x)) + ((47.0 / 7.0) * f1(x)) + (-(59.0 / 66.0) * f6(x)) + (-(30.0 / 57.0) * f0(x)) + (-(10.0 / 78.0) * f0(x)) + ((79.0 / 77.0) * f3(x)) + ((29.0 / 81.0) * f4(x)) + (-(69.0 / 72.0) * f3(x)) + ((56.0 / 5.0) * f3(x)) + (-(43.0 / 93.0) * f1(x)) + (-(2.0 / 46.0) * f2(x)) + ((74.0 / 51.0) * f6(x)) + (-(85.0 / 87.0) * f4(x)) + (-(77.0 / 12.0) * f0(x)) + ((86.0 / 69.0) * f5(x)) + (-(45.0 / 44.0) * f1(x)) + (-(15.0 / 33.0) * f7(x)) + (-(63.0 / 45.0) * f3(x)) + (-(41.0 / 92.0) * f6(x)) + ((91.0 / 91.0) * f7(x)) + (-(17.0 / 12.0) * f0(x)) + (-(83.0 / 36.0) * f0(x)) + ((77.0 / 46.0) * f6(x)) + ((49.0 / 86.0) * f3(x)) + (-(82.0 / 88.0) * f2(x)) + (-(33.0 / 88.0) * f7(x)) + (-(24.0 / 67.0) * f3(x)) + (-(71.0 / 97.0) * f5(x)) + ((15.0 / 13.0) * f5(x)) + ((47.0 / 43.0) * f4(x)) + (-(61.0 / 99.0) * f0(x)) + ((63.0 / 69.0) * f5(x)) + ((72.0 / 38.0) * f4(x)) + (-(40.0 / 38.0) * f4(x)) + ((4.0 / 77.0) * f7(x)) + (-(34.0 / 97.0) * f4(x)) + ((43.0 / 97.0) * f4(x)) + (-(60.0 / 19.0) * f1(x)) + ((15.0 / 3.0) * f1(x)) + ((17.0 / 66.0) * f6(x)) + ((59.0 / 57.0) * f0(x)) + (54.0 * f5(x)) + ((8.0 / 2.0) * f5(x)) + ((59.0 / 85.0) * f0(x)) + (-(57.0 / 38.0) * f7(x)) + ((42.0 / 3.0) * f7(x)) + ((52.0 / 48.0) * f0(x)) + (-(86.0 / 10.0) * f7(x)) + ((78.0 / 63.0) * f1(x)) + (-(11.0 / 59.0) * f2(x)) + ((57.0 / 29.0) * f3(x)) + (-(34.0 / 35.0) * f1(x)) + ((18.0 / 84.0) * f2(x)) + (-(68.0 / 22.0) * f4(x)) + (-(95.0 / 56.0) * f6(x)) + ((34.0 / 43.0) * f7(x)) + ((73.0 / 85.0) * f2(x)) + ((98.0 / 69.0) * f7(x)) + ((3.0 / 6.0) * f7(x)) + ((84.0 / 92.0) * f5(x)) + (-(41.0 / 81.0) * f4(x)) + (-(98.0 / 50.0) * f1(x)) + (-(76.0 / 30.0) * f0(x)) + (-(47.0 / 32.0) * f2(x)) + ((59.0 / 6.0) * f3(x)) + ((26.0 / 14.0) * f2(x)) + (-(58.0 / 40.0) * f0(x)) + ((61.0 / 28.0) * f7(x))) - (((52.0 / 80.0) * l6) + (-(20.0 / 53.0) * l0) + ((14.0 / 81.0) * l2) + ((61.0 / 5.0) * l2) + (-(96.0 / 63.0) * l0) + (-(19.0 / 27.0) * l2) + (-(30.0 / 40.0) * l6) + (-(12.0 / 84.0) * l7) + (-(83.0 / 100.0) * l1) + ((67.0 / 99.0) * l0) + ((83.0 / 23.0) * l5) + (-(45.0 / 32.0) * l5) + (-(83.0 / 100.0) * l5) + ((43.0 / 26.0) * l5) + ((33.0 / 30.0) * l6) + (-(67.0 / 74.0) * l6) + ((95.0 / 60.0) * l6) + (-(79.0 / 46.0) * l1) + ((76.0 / 75.0) * l3) + (-(82.0 / 18.0) * l7) + (-(14.0 / 6.0) * l0) + (-(100.0 / 57.0) * l2) + ((89.0 / 58.0) * l0) + (-(32.0 / 28.0) * l6) + (-(23.0 / 38.0) * l3) + ((61.0 / 76.0) * l3) + (-(97.0 / 25.0) * l7) + (-(52.0 / 85.0) * l7) + ((61.0 / 58.0) * l4) + (-(41.0 / 99.0) * l0) + (-(100.0 / 8.0) * l3) + (-(71.0 / 13.0) * l3) + (18.0 * l6) + (-(25.0 / 40.0) * l7) + (-(1.0 / 53.0) * l4) + (-(84.0 / 37.0) * l4) + (-(57.0 / 87.0) * l0) + (-(9.0 / 98.0) * l4) + ((34.0 / 9.0) * l3) + ((30.0 / 78.0) * l7) + ((84.0 / 55.0) * l2) + (-(23.0 / 95.0) * l1) + ((63.0 / 84.0) * l6) + (-(6.0 / 37.0) * l0) + (-(52.0 / 88.0) * l7) + (-(33.0 / 14.0) * l5) + ((7.0 / 39.0) * l2) + (-(18.0 / 96.0) * l3) + (-(0.0 / 20.0) * l4) + (-(13.0 / 42.0) * l2) + (-(12.0 / 5.0) * l2) + ((32.0 / 24.0) * l1) + ((40.0 / 79.0) * l7) + ((12.0 / 36.0) * l4) + (-(69.0 / 45.0) * l7) + (-(5.0 / 93.0) * l5) + (-(3.0 / 73.0) * l0) + ((92.0 / 83.0) * l2) + (-(4.0 / 43.0) * l2) + (-(1.0 / 70.0) * l5) + (-(58.0 / 87.0) * l0) + ((47.0 / 7.0) * l1) + (-(59.0 / 66.0) * l6) + (-(30.0 / 57.0) * l0) + (-(10.0 / 78.0) * l0) + ((79.0 / 77.0) * l3) + ((29.0 / 81.0) * l4) + (-(69.0 / 72.0) * l3) + ((56.0 / 5.0) * l3) + (-(43.0 / 93.0) * l1) + (-(2.0 / 46.0) * l2) + ((74.0 / 51.0) * l6) + (-(85.0 / 87.0) * l4) + (-(77.0 / 12.0) * l0) + ((86.0 / 69.0) * l5) + (-(45.0 / 44.0) * l1) + (-(15.0 / 33.0) * l7) + (-(63.0 / 45.0) * l3) + (-(41.0 / 92.0) * l6) + ((91.0 / 91.0) * l7) + (-(17.0 / 12.0) * l0) + (-(83.0 / 36.0) * l0) + ((77.0 / 46.0) * l6) + ((49.0 / 86.0) * l3) + (-(82.0 / 88.0) * l2) + (-(33.0 / 88.0) * l7) + (-(24.0 / 67.0) * l3) + (-(71.0 / 97.0) * l5) + ((15.0 / 13.0) * l5) + ((47.0 / 43.0) * l4) + (-(61.0 / 99.0) * l0) + ((63.0 / 69.0) * l5) + ((72.0 / 38.0) * l4) + (-(40.0 / 38.0) * l4) + ((4.0 / 77.0) * l7) + (-(34.0 / 97.0) * l4) + ((43.0 / 97.0) * l4) + (-(60.0 / 19.0) * l1) + ((15.0 / 3.0) * l1) + ((17.0 / 66.0) * l6) + ((59.0 / 57.0) * l0) + (54.0 * l5) + ((8.0 / 2.0) * l5) + ((59.0 / 85.0) * l0) + (-(57.0 / 38.0) * l7) + ((42.0 / 3.0) * l7) + ((52.0 / 48.0) * l0) + (-(86.0 / 10.0) * l7) + ((78.0 / 63.0) * l1) + (-(11.0 / 59.0) * l2) + ((57.0 / 29.0) * l3) + (-(34.0 / 35.0) * l1) + ((18.0 / 84.0) * l2) + (-(68.0 / 22.0) * l4) + (-(95.0 / 56.0) * l6) + ((34.0 / 43.0) * l7) + ((73.0 / 85.0) * l2) + ((98.0 / 69.0) * l7) + ((3.0 / 6.0) * l7) + ((84.0 / 92.0) * l5) + (-(41.0 / 81.0) * l4) + (-(98.0 / 50.0) * l1) + (-(76.0 / 30.0) * l0) + (-(47.0 / 32.0) * l2) + ((59.0 / 6.0) * l3) + ((26.0 / 14.0) * l2) + (-(58.0 / 40.0) * l0) + ((61.0 / 28.0) * l7)))) < epsilon))))
tff(formula_9,conjecture,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != a )
& ( $greatereq($difference(X,a),0.0)
=> $less($difference(X,a),Delta) )
& ( ~ $greatereq($difference(X,a),0.0)
=> $less($uminus($difference(X,a)),Delta) ) )
=> ( ( 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%------------------------------------------------------------------------------