TPTP Problem File: ANA134_1.002.016.p
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%------------------------------------------------------------------------------
% File : ANA134_1.002.016 : TPTP v8.2.0. Released v8.2.0.
% Domain : Number theory
% Problem : composition_tower_f2_sz16__1
% Version : Especial.
% English : lim[x -> l1](f_i1(x)) ~ l
% lim[x -> l2](f_i2(x)) ~ l1
% lim[x -> l3](f_i3(x)) ~ l2
% ...
% lim[x -> l15](f_i15(x)) ~ l14
% lim[x -> a](f_i16(x)) ~ l15
% ============================
% lim[x -> a](f_i1(f_i2(f_i3(...f_i16(x)...)))) ~ l
% where
% - 2 functions with arity 1
% - 16 number of function applications
% Refs : [Sch22] Schoisswohl (2022), Email to G. Sutcliffe
% : [KK+23] Korovin et al. (2023), ALASCA: Reasoning in Quantified
% Source : [Sch22]
% Names : composition_tower_f2_sz16__1.smt2 [Sch22]
% Status : Theorem
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 36 ( 0 unt; 19 typ; 0 def)
% Number of atoms : 187 ( 17 equ)
% Maximal formula atoms : 11 ( 11 avg)
% Number of connectives : 221 ( 51 ~; 0 |; 68 &)
% ( 0 <=>; 102 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 11 avg)
% Maximal term depth : 19 ( 2 avg)
% Number arithmetic : 493 ( 170 atm; 170 fun; 102 num; 51 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 : 22 ( 19 usr; 18 con; 0-2 aty)
% Number of variables : 51 ( 34 !; 17 ?; 51 :)
% SPC : TF0_THM_EQU_ARI
% Comments : Translated from SMT UFLRA by SMTtoTPTP.
%------------------------------------------------------------------------------
%% Declarations:
tff(l14,type,
l14: $real ).
tff(l4,type,
l4: $real ).
tff(l5,type,
l5: $real ).
tff(f0,type,
f0: $real > $real ).
tff(l2,type,
l2: $real ).
tff(a,type,
a: $real ).
tff(l15,type,
l15: $real ).
tff(f1,type,
f1: $real > $real ).
tff(l10,type,
l10: $real ).
tff(l1,type,
l1: $real ).
tff(l13,type,
l13: $real ).
tff(l11,type,
l11: $real ).
tff(l7,type,
l7: $real ).
tff(l,type,
l: $real ).
tff(l9,type,
l9: $real ).
tff(l3,type,
l3: $real ).
tff(l12,type,
l12: $real ).
tff(l8,type,
l8: $real ).
tff(l6,type,
l6: $real ).
%% Assertions:
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l1) ∧ ((if ((x - l1) ≥ 0.0) (x - l1) else -(x - l1)) < delta)) ⇒ ((if ((f1(x) - l) ≥ 0.0) (f1(x) - l) else -(f1(x) - l)) < epsilon))))
tff(formula_1,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l1 )
& ( $greatereq($difference(X,l1),0.0)
=> $less($difference(X,l1),Delta) )
& ( ~ $greatereq($difference(X,l1),0.0)
=> $less($uminus($difference(X,l1)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l),0.0)
=> $less($difference(f1(X),l),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l),0.0)
=> $less($uminus($difference(f1(X),l)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l2) ∧ ((if ((x - l2) ≥ 0.0) (x - l2) else -(x - l2)) < 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 != l2 )
& ( $greatereq($difference(X,l2),0.0)
=> $less($difference(X,l2),Delta) )
& ( ~ $greatereq($difference(X,l2),0.0)
=> $less($uminus($difference(X,l2)),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 = l3) ∧ ((if ((x - l3) ≥ 0.0) (x - l3) else -(x - l3)) < delta)) ⇒ ((if ((f0(x) - l2) ≥ 0.0) (f0(x) - l2) else -(f0(x) - l2)) < epsilon))))
tff(formula_3,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l3 )
& ( $greatereq($difference(X,l3),0.0)
=> $less($difference(X,l3),Delta) )
& ( ~ $greatereq($difference(X,l3),0.0)
=> $less($uminus($difference(X,l3)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l2),0.0)
=> $less($difference(f0(X),l2),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l2),0.0)
=> $less($uminus($difference(f0(X),l2)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l4) ∧ ((if ((x - l4) ≥ 0.0) (x - l4) else -(x - l4)) < delta)) ⇒ ((if ((f0(x) - l3) ≥ 0.0) (f0(x) - l3) else -(f0(x) - l3)) < epsilon))))
tff(formula_4,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l4 )
& ( $greatereq($difference(X,l4),0.0)
=> $less($difference(X,l4),Delta) )
& ( ~ $greatereq($difference(X,l4),0.0)
=> $less($uminus($difference(X,l4)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l3),0.0)
=> $less($difference(f0(X),l3),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l3),0.0)
=> $less($uminus($difference(f0(X),l3)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l5) ∧ ((if ((x - l5) ≥ 0.0) (x - l5) else -(x - l5)) < delta)) ⇒ ((if ((f0(x) - l4) ≥ 0.0) (f0(x) - l4) else -(f0(x) - l4)) < epsilon))))
tff(formula_5,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l5 )
& ( $greatereq($difference(X,l5),0.0)
=> $less($difference(X,l5),Delta) )
& ( ~ $greatereq($difference(X,l5),0.0)
=> $less($uminus($difference(X,l5)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l4),0.0)
=> $less($difference(f0(X),l4),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l4),0.0)
=> $less($uminus($difference(f0(X),l4)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l6) ∧ ((if ((x - l6) ≥ 0.0) (x - l6) else -(x - l6)) < delta)) ⇒ ((if ((f1(x) - l5) ≥ 0.0) (f1(x) - l5) else -(f1(x) - l5)) < epsilon))))
tff(formula_6,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l6 )
& ( $greatereq($difference(X,l6),0.0)
=> $less($difference(X,l6),Delta) )
& ( ~ $greatereq($difference(X,l6),0.0)
=> $less($uminus($difference(X,l6)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l5),0.0)
=> $less($difference(f1(X),l5),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l5),0.0)
=> $less($uminus($difference(f1(X),l5)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l7) ∧ ((if ((x - l7) ≥ 0.0) (x - l7) else -(x - l7)) < delta)) ⇒ ((if ((f0(x) - l6) ≥ 0.0) (f0(x) - l6) else -(f0(x) - l6)) < epsilon))))
tff(formula_7,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l7 )
& ( $greatereq($difference(X,l7),0.0)
=> $less($difference(X,l7),Delta) )
& ( ~ $greatereq($difference(X,l7),0.0)
=> $less($uminus($difference(X,l7)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l6),0.0)
=> $less($difference(f0(X),l6),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l6),0.0)
=> $less($uminus($difference(f0(X),l6)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l8) ∧ ((if ((x - l8) ≥ 0.0) (x - l8) else -(x - l8)) < delta)) ⇒ ((if ((f1(x) - l7) ≥ 0.0) (f1(x) - l7) else -(f1(x) - l7)) < epsilon))))
tff(formula_8,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l8 )
& ( $greatereq($difference(X,l8),0.0)
=> $less($difference(X,l8),Delta) )
& ( ~ $greatereq($difference(X,l8),0.0)
=> $less($uminus($difference(X,l8)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l7),0.0)
=> $less($difference(f1(X),l7),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l7),0.0)
=> $less($uminus($difference(f1(X),l7)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l9) ∧ ((if ((x - l9) ≥ 0.0) (x - l9) else -(x - l9)) < delta)) ⇒ ((if ((f1(x) - l8) ≥ 0.0) (f1(x) - l8) else -(f1(x) - l8)) < epsilon))))
tff(formula_9,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l9 )
& ( $greatereq($difference(X,l9),0.0)
=> $less($difference(X,l9),Delta) )
& ( ~ $greatereq($difference(X,l9),0.0)
=> $less($uminus($difference(X,l9)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l8),0.0)
=> $less($difference(f1(X),l8),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l8),0.0)
=> $less($uminus($difference(f1(X),l8)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l10) ∧ ((if ((x - l10) ≥ 0.0) (x - l10) else -(x - l10)) < delta)) ⇒ ((if ((f0(x) - l9) ≥ 0.0) (f0(x) - l9) else -(f0(x) - l9)) < epsilon))))
tff(formula_10,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l10 )
& ( $greatereq($difference(X,l10),0.0)
=> $less($difference(X,l10),Delta) )
& ( ~ $greatereq($difference(X,l10),0.0)
=> $less($uminus($difference(X,l10)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l9),0.0)
=> $less($difference(f0(X),l9),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l9),0.0)
=> $less($uminus($difference(f0(X),l9)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l11) ∧ ((if ((x - l11) ≥ 0.0) (x - l11) else -(x - l11)) < delta)) ⇒ ((if ((f0(x) - l10) ≥ 0.0) (f0(x) - l10) else -(f0(x) - l10)) < epsilon))))
tff(formula_11,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l11 )
& ( $greatereq($difference(X,l11),0.0)
=> $less($difference(X,l11),Delta) )
& ( ~ $greatereq($difference(X,l11),0.0)
=> $less($uminus($difference(X,l11)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l10),0.0)
=> $less($difference(f0(X),l10),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l10),0.0)
=> $less($uminus($difference(f0(X),l10)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l12) ∧ ((if ((x - l12) ≥ 0.0) (x - l12) else -(x - l12)) < delta)) ⇒ ((if ((f0(x) - l11) ≥ 0.0) (f0(x) - l11) else -(f0(x) - l11)) < epsilon))))
tff(formula_12,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l12 )
& ( $greatereq($difference(X,l12),0.0)
=> $less($difference(X,l12),Delta) )
& ( ~ $greatereq($difference(X,l12),0.0)
=> $less($uminus($difference(X,l12)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l11),0.0)
=> $less($difference(f0(X),l11),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l11),0.0)
=> $less($uminus($difference(f0(X),l11)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l13) ∧ ((if ((x - l13) ≥ 0.0) (x - l13) else -(x - l13)) < delta)) ⇒ ((if ((f1(x) - l12) ≥ 0.0) (f1(x) - l12) else -(f1(x) - l12)) < epsilon))))
tff(formula_13,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l13 )
& ( $greatereq($difference(X,l13),0.0)
=> $less($difference(X,l13),Delta) )
& ( ~ $greatereq($difference(X,l13),0.0)
=> $less($uminus($difference(X,l13)),Delta) ) )
=> ( ( $greatereq($difference(f1(X),l12),0.0)
=> $less($difference(f1(X),l12),Epsilon) )
& ( ~ $greatereq($difference(f1(X),l12),0.0)
=> $less($uminus($difference(f1(X),l12)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l14) ∧ ((if ((x - l14) ≥ 0.0) (x - l14) else -(x - l14)) < delta)) ⇒ ((if ((f0(x) - l13) ≥ 0.0) (f0(x) - l13) else -(f0(x) - l13)) < epsilon))))
tff(formula_14,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l14 )
& ( $greatereq($difference(X,l14),0.0)
=> $less($difference(X,l14),Delta) )
& ( ~ $greatereq($difference(X,l14),0.0)
=> $less($uminus($difference(X,l14)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l13),0.0)
=> $less($difference(f0(X),l13),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l13),0.0)
=> $less($uminus($difference(f0(X),l13)),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ x:Real ((¬(x = l15) ∧ ((if ((x - l15) ≥ 0.0) (x - l15) else -(x - l15)) < delta)) ⇒ ((if ((f0(x) - l14) ≥ 0.0) (f0(x) - l14) else -(f0(x) - l14)) < epsilon))))
tff(formula_15,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [X: $real] :
( ( ( X != l15 )
& ( $greatereq($difference(X,l15),0.0)
=> $less($difference(X,l15),Delta) )
& ( ~ $greatereq($difference(X,l15),0.0)
=> $less($uminus($difference(X,l15)),Delta) ) )
=> ( ( $greatereq($difference(f0(X),l14),0.0)
=> $less($difference(f0(X),l14),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l14),0.0)
=> $less($uminus($difference(f0(X),l14)),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 ((f0(x) - l15) ≥ 0.0) (f0(x) - l15) else -(f0(x) - l15)) < epsilon))))
tff(formula_16,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),l15),0.0)
=> $less($difference(f0(X),l15),Epsilon) )
& ( ~ $greatereq($difference(f0(X),l15),0.0)
=> $less($uminus($difference(f0(X),l15)),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(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(x)))))))))))))))) - l) ≥ 0.0) (f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(x)))))))))))))))) - l) else -(f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(x)))))))))))))))) - l)) < epsilon))))
tff(formula_17,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(f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(X)))))))))))))))),l),0.0)
=> $less($difference(f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(X)))))))))))))))),l),Epsilon) )
& ( ~ $greatereq($difference(f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(X)))))))))))))))),l),0.0)
=> $less($uminus($difference(f1(f1(f0(f0(f0(f1(f0(f1(f1(f0(f0(f0(f1(f0(f0(f0(X)))))))))))))))),l)),Epsilon) ) ) ) ) ) ).
%------------------------------------------------------------------------------