TPTP Problem File: ANA143_1.002.016.p
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%------------------------------------------------------------------------------
% File : ANA143_1.002.016 : TPTP v8.2.0. Released v8.2.0.
% Domain : Number theory
% Problem : linear_limit_n2_m16__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 = 2
% m = 16
% 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_n2_m16__1.smt2 [Sch22]
% Status : Theorem
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 8 ( 0 unt; 5 typ; 0 def)
% Number of atoms : 33 ( 3 equ)
% Maximal formula atoms : 11 ( 11 avg)
% Number of connectives : 39 ( 9 ~; 0 |; 12 &)
% ( 0 <=>; 18 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 11 avg)
% Maximal term depth : 21 ( 3 avg)
% Number arithmetic : 751 ( 30 atm; 446 fun; 266 num; 9 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 40 ( 5 usr; 33 con; 0-2 aty)
% Number of variables : 9 ( 6 !; 3 ?; 9 :)
% SPC : TF0_THM_EQU_ARI
% Comments : Translated from SMT UFLRA by SMTtoTPTP.
%------------------------------------------------------------------------------
%% Declarations:
tff(f0,type,
f0: $real > $real ).
tff(l0,type,
l0: $real ).
tff(a,type,
a: $real ).
tff(f1,type,
f1: $real > $real ).
tff(l1,type,
l1: $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 (((((52.0 / 80.0) * f0(x)) + ((20.0 / 39.0) * f0(x)) + ((14.0 / 81.0) * f0(x)) + ((89.0 / 40.0) * f0(x)) + (-(24.0 / 32.0) * f1(x)) + ((12.0 / 64.0) * f0(x)) + (-(2.0 / 93.0) * f0(x)) + (-(27.0 / 17.0) * f1(x)) + ((38.0 / 100.0) * f0(x)) + (-72.0 * f1(x)) + ((13.0 / 87.0) * f1(x)) + ((16.0 / 6.0) * f1(x)) + ((80.0 / 65.0) * f0(x)) + (-(63.0 / 33.0) * f0(x)) + ((84.0 / 64.0) * f1(x)) + (-(68.0 / 22.0) * f1(x))) - (((52.0 / 80.0) * l0) + ((20.0 / 39.0) * l0) + ((14.0 / 81.0) * l0) + ((89.0 / 40.0) * l0) + (-(24.0 / 32.0) * l1) + ((12.0 / 64.0) * l0) + (-(2.0 / 93.0) * l0) + (-(27.0 / 17.0) * l1) + ((38.0 / 100.0) * l0) + (-72.0 * l1) + ((13.0 / 87.0) * l1) + ((16.0 / 6.0) * l1) + ((80.0 / 65.0) * l0) + (-(63.0 / 33.0) * l0) + ((84.0 / 64.0) * l1) + (-(68.0 / 22.0) * l1))) ≥ 0.0) ((((52.0 / 80.0) * f0(x)) + ((20.0 / 39.0) * f0(x)) + ((14.0 / 81.0) * f0(x)) + ((89.0 / 40.0) * f0(x)) + (-(24.0 / 32.0) * f1(x)) + ((12.0 / 64.0) * f0(x)) + (-(2.0 / 93.0) * f0(x)) + (-(27.0 / 17.0) * f1(x)) + ((38.0 / 100.0) * f0(x)) + (-72.0 * f1(x)) + ((13.0 / 87.0) * f1(x)) + ((16.0 / 6.0) * f1(x)) + ((80.0 / 65.0) * f0(x)) + (-(63.0 / 33.0) * f0(x)) + ((84.0 / 64.0) * f1(x)) + (-(68.0 / 22.0) * f1(x))) - (((52.0 / 80.0) * l0) + ((20.0 / 39.0) * l0) + ((14.0 / 81.0) * l0) + ((89.0 / 40.0) * l0) + (-(24.0 / 32.0) * l1) + ((12.0 / 64.0) * l0) + (-(2.0 / 93.0) * l0) + (-(27.0 / 17.0) * l1) + ((38.0 / 100.0) * l0) + (-72.0 * l1) + ((13.0 / 87.0) * l1) + ((16.0 / 6.0) * l1) + ((80.0 / 65.0) * l0) + (-(63.0 / 33.0) * l0) + ((84.0 / 64.0) * l1) + (-(68.0 / 22.0) * l1))) else -((((52.0 / 80.0) * f0(x)) + ((20.0 / 39.0) * f0(x)) + ((14.0 / 81.0) * f0(x)) + ((89.0 / 40.0) * f0(x)) + (-(24.0 / 32.0) * f1(x)) + ((12.0 / 64.0) * f0(x)) + (-(2.0 / 93.0) * f0(x)) + (-(27.0 / 17.0) * f1(x)) + ((38.0 / 100.0) * f0(x)) + (-72.0 * f1(x)) + ((13.0 / 87.0) * f1(x)) + ((16.0 / 6.0) * f1(x)) + ((80.0 / 65.0) * f0(x)) + (-(63.0 / 33.0) * f0(x)) + ((84.0 / 64.0) * f1(x)) + (-(68.0 / 22.0) * f1(x))) - (((52.0 / 80.0) * l0) + ((20.0 / 39.0) * l0) + ((14.0 / 81.0) * l0) + ((89.0 / 40.0) * l0) + (-(24.0 / 32.0) * l1) + ((12.0 / 64.0) * l0) + (-(2.0 / 93.0) * l0) + (-(27.0 / 17.0) * l1) + ((38.0 / 100.0) * l0) + (-72.0 * l1) + ((13.0 / 87.0) * l1) + ((16.0 / 6.0) * l1) + ((80.0 / 65.0) * l0) + (-(63.0 / 33.0) * l0) + ((84.0 / 64.0) * l1) + (-(68.0 / 22.0) * l1)))) < epsilon))))
tff(formula_3,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) ) )
=> ( ( $greatereq($difference($sum($product($quotient(52.0,80.0),f0(X)),$sum($product($quotient(20.0,39.0),f0(X)),$sum($product($quotient(14.0,81.0),f0(X)),$sum($product($quotient(89.0,40.0),f0(X)),$sum($product($uminus($quotient(24.0,32.0)),f1(X)),$sum($product($quotient(12.0,64.0),f0(X)),$sum($product($uminus($quotient(2.0,93.0)),f0(X)),$sum($product($uminus($quotient(27.0,17.0)),f1(X)),$sum($product($quotient(38.0,100.0),f0(X)),$sum($product($uminus(72.0),f1(X)),$sum($product($quotient(13.0,87.0),f1(X)),$sum($product($quotient(16.0,6.0),f1(X)),$sum($product($quotient(80.0,65.0),f0(X)),$sum($product($uminus($quotient(63.0,33.0)),f0(X)),$sum($product($quotient(84.0,64.0),f1(X)),$product($uminus($quotient(68.0,22.0)),f1(X))))))))))))))))),$sum($product($quotient(52.0,80.0),l0),$sum($product($quotient(20.0,39.0),l0),$sum($product($quotient(14.0,81.0),l0),$sum($product($quotient(89.0,40.0),l0),$sum($product($uminus($quotient(24.0,32.0)),l1),$sum($product($quotient(12.0,64.0),l0),$sum($product($uminus($quotient(2.0,93.0)),l0),$sum($product($uminus($quotient(27.0,17.0)),l1),$sum($product($quotient(38.0,100.0),l0),$sum($product($uminus(72.0),l1),$sum($product($quotient(13.0,87.0),l1),$sum($product($quotient(16.0,6.0),l1),$sum($product($quotient(80.0,65.0),l0),$sum($product($uminus($quotient(63.0,33.0)),l0),$sum($product($quotient(84.0,64.0),l1),$product($uminus($quotient(68.0,22.0)),l1))))))))))))))))),0.0)
=> $less($difference($sum($product($quotient(52.0,80.0),f0(X)),$sum($product($quotient(20.0,39.0),f0(X)),$sum($product($quotient(14.0,81.0),f0(X)),$sum($product($quotient(89.0,40.0),f0(X)),$sum($product($uminus($quotient(24.0,32.0)),f1(X)),$sum($product($quotient(12.0,64.0),f0(X)),$sum($product($uminus($quotient(2.0,93.0)),f0(X)),$sum($product($uminus($quotient(27.0,17.0)),f1(X)),$sum($product($quotient(38.0,100.0),f0(X)),$sum($product($uminus(72.0),f1(X)),$sum($product($quotient(13.0,87.0),f1(X)),$sum($product($quotient(16.0,6.0),f1(X)),$sum($product($quotient(80.0,65.0),f0(X)),$sum($product($uminus($quotient(63.0,33.0)),f0(X)),$sum($product($quotient(84.0,64.0),f1(X)),$product($uminus($quotient(68.0,22.0)),f1(X))))))))))))))))),$sum($product($quotient(52.0,80.0),l0),$sum($product($quotient(20.0,39.0),l0),$sum($product($quotient(14.0,81.0),l0),$sum($product($quotient(89.0,40.0),l0),$sum($product($uminus($quotient(24.0,32.0)),l1),$sum($product($quotient(12.0,64.0),l0),$sum($product($uminus($quotient(2.0,93.0)),l0),$sum($product($uminus($quotient(27.0,17.0)),l1),$sum($product($quotient(38.0,100.0),l0),$sum($product($uminus(72.0),l1),$sum($product($quotient(13.0,87.0),l1),$sum($product($quotient(16.0,6.0),l1),$sum($product($quotient(80.0,65.0),l0),$sum($product($uminus($quotient(63.0,33.0)),l0),$sum($product($quotient(84.0,64.0),l1),$product($uminus($quotient(68.0,22.0)),l1))))))))))))))))),Epsilon) )
& ( ~ $greatereq($difference($sum($product($quotient(52.0,80.0),f0(X)),$sum($product($quotient(20.0,39.0),f0(X)),$sum($product($quotient(14.0,81.0),f0(X)),$sum($product($quotient(89.0,40.0),f0(X)),$sum($product($uminus($quotient(24.0,32.0)),f1(X)),$sum($product($quotient(12.0,64.0),f0(X)),$sum($product($uminus($quotient(2.0,93.0)),f0(X)),$sum($product($uminus($quotient(27.0,17.0)),f1(X)),$sum($product($quotient(38.0,100.0),f0(X)),$sum($product($uminus(72.0),f1(X)),$sum($product($quotient(13.0,87.0),f1(X)),$sum($product($quotient(16.0,6.0),f1(X)),$sum($product($quotient(80.0,65.0),f0(X)),$sum($product($uminus($quotient(63.0,33.0)),f0(X)),$sum($product($quotient(84.0,64.0),f1(X)),$product($uminus($quotient(68.0,22.0)),f1(X))))))))))))))))),$sum($product($quotient(52.0,80.0),l0),$sum($product($quotient(20.0,39.0),l0),$sum($product($quotient(14.0,81.0),l0),$sum($product($quotient(89.0,40.0),l0),$sum($product($uminus($quotient(24.0,32.0)),l1),$sum($product($quotient(12.0,64.0),l0),$sum($product($uminus($quotient(2.0,93.0)),l0),$sum($product($uminus($quotient(27.0,17.0)),l1),$sum($product($quotient(38.0,100.0),l0),$sum($product($uminus(72.0),l1),$sum($product($quotient(13.0,87.0),l1),$sum($product($quotient(16.0,6.0),l1),$sum($product($quotient(80.0,65.0),l0),$sum($product($uminus($quotient(63.0,33.0)),l0),$sum($product($quotient(84.0,64.0),l1),$product($uminus($quotient(68.0,22.0)),l1))))))))))))))))),0.0)
=> $less($uminus($difference($sum($product($quotient(52.0,80.0),f0(X)),$sum($product($quotient(20.0,39.0),f0(X)),$sum($product($quotient(14.0,81.0),f0(X)),$sum($product($quotient(89.0,40.0),f0(X)),$sum($product($uminus($quotient(24.0,32.0)),f1(X)),$sum($product($quotient(12.0,64.0),f0(X)),$sum($product($uminus($quotient(2.0,93.0)),f0(X)),$sum($product($uminus($quotient(27.0,17.0)),f1(X)),$sum($product($quotient(38.0,100.0),f0(X)),$sum($product($uminus(72.0),f1(X)),$sum($product($quotient(13.0,87.0),f1(X)),$sum($product($quotient(16.0,6.0),f1(X)),$sum($product($quotient(80.0,65.0),f0(X)),$sum($product($uminus($quotient(63.0,33.0)),f0(X)),$sum($product($quotient(84.0,64.0),f1(X)),$product($uminus($quotient(68.0,22.0)),f1(X))))))))))))))))),$sum($product($quotient(52.0,80.0),l0),$sum($product($quotient(20.0,39.0),l0),$sum($product($quotient(14.0,81.0),l0),$sum($product($quotient(89.0,40.0),l0),$sum($product($uminus($quotient(24.0,32.0)),l1),$sum($product($quotient(12.0,64.0),l0),$sum($product($uminus($quotient(2.0,93.0)),l0),$sum($product($uminus($quotient(27.0,17.0)),l1),$sum($product($quotient(38.0,100.0),l0),$sum($product($uminus(72.0),l1),$sum($product($quotient(13.0,87.0),l1),$sum($product($quotient(16.0,6.0),l1),$sum($product($quotient(80.0,65.0),l0),$sum($product($uminus($quotient(63.0,33.0)),l0),$sum($product($quotient(84.0,64.0),l1),$product($uminus($quotient(68.0,22.0)),l1)))))))))))))))))),Epsilon) ) ) ) ) ) ).
%------------------------------------------------------------------------------