TPTP Problem File: ANA141_1.p
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%------------------------------------------------------------------------------
% File : ANA141_1 : TPTP v9.0.0. Released v8.2.0.
% Domain : Number theory
% Problem : crafted_lim_fxy
% Version : Especial.
% English : lim[x -> a](lim[y -> b])(f(x, y)) = lim[y -> b](lim[x -> a])(f(x, y))
% Refs : [Sch22] Schoisswohl (2022), Email to G. Sutcliffe
% : [KK+23] Korovin et al. (2023), ALASCA: Reasoning in Quantified
% Source : [Sch22]
% Names : crafted_lim_fxy.smt2 [Sch22]
% Status : Theorem
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 12 ( 1 unt; 7 typ; 0 def)
% Number of atoms : 45 ( 5 equ)
% Maximal formula atoms : 11 ( 9 avg)
% Number of connectives : 52 ( 12 ~; 0 |; 16 &)
% ( 0 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 118 ( 40 atm; 40 fun; 24 num; 14 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 4 ( 3 >; 1 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 10 ( 7 usr; 5 con; 0-2 aty)
% Number of variables : 14 ( 10 !; 4 ?; 14 :)
% SPC : TF0_THM_EQU_ARI
% Comments : Translated from SMT UFLRA by SMTtoTPTP.
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%% Declarations:
tff(f,type,
f: ( $real * $real ) > $real ).
tff(fa,type,
fa: $real > $real ).
tff(lxy,type,
lxy: $real ).
tff(a,type,
a: $real ).
tff(b,type,
b: $real ).
tff(lyx,type,
lyx: $real ).
tff(fb,type,
fb: $real > $real ).
%% Assertions:
%% ∀ y:Real ∀ 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 ((f(x, y) - fa(y)) ≥ 0.0) (f(x, y) - fa(y)) else -(f(x, y) - fa(y))) < epsilon))))
tff(formula_1,axiom,
! [Y: $real,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(f(X,Y),fa(Y)),0.0)
=> $less($difference(f(X,Y),fa(Y)),Epsilon) )
& ( ~ $greatereq($difference(f(X,Y),fa(Y)),0.0)
=> $less($uminus($difference(f(X,Y),fa(Y))),Epsilon) ) ) ) ) ) ).
%% ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ y:Real ((¬(y = b) ∧ ((if ((y - b) ≥ 0.0) (y - b) else -(y - b)) < delta)) ⇒ ((if ((fa(y) - lxy) ≥ 0.0) (fa(y) - lxy) else -(fa(y) - lxy)) < epsilon))))
tff(formula_2,axiom,
! [Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [Y: $real] :
( ( ( Y != b )
& ( $greatereq($difference(Y,b),0.0)
=> $less($difference(Y,b),Delta) )
& ( ~ $greatereq($difference(Y,b),0.0)
=> $less($uminus($difference(Y,b)),Delta) ) )
=> ( ( $greatereq($difference(fa(Y),lxy),0.0)
=> $less($difference(fa(Y),lxy),Epsilon) )
& ( ~ $greatereq($difference(fa(Y),lxy),0.0)
=> $less($uminus($difference(fa(Y),lxy)),Epsilon) ) ) ) ) ) ).
%% ∀ x:Real ∀ epsilon:Real ((0.0 < epsilon) ⇒ ∃ delta:Real ((0.0 < delta) ∧ ∀ y:Real ((¬(y = b) ∧ ((if ((y - b) ≥ 0.0) (y - b) else -(y - b)) < delta)) ⇒ ((if ((f(x, y) - fb(x)) ≥ 0.0) (f(x, y) - fb(x)) else -(f(x, y) - fb(x))) < epsilon))))
tff(formula_3,axiom,
! [X: $real,Epsilon: $real] :
( $less(0.0,Epsilon)
=> ? [Delta: $real] :
( $less(0.0,Delta)
& ! [Y: $real] :
( ( ( Y != b )
& ( $greatereq($difference(Y,b),0.0)
=> $less($difference(Y,b),Delta) )
& ( ~ $greatereq($difference(Y,b),0.0)
=> $less($uminus($difference(Y,b)),Delta) ) )
=> ( ( $greatereq($difference(f(X,Y),fb(X)),0.0)
=> $less($difference(f(X,Y),fb(X)),Epsilon) )
& ( ~ $greatereq($difference(f(X,Y),fb(X)),0.0)
=> $less($uminus($difference(f(X,Y),fb(X))),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 ((fb(x) - lyx) ≥ 0.0) (fb(x) - lyx) else -(fb(x) - lyx)) < 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(fb(X),lyx),0.0)
=> $less($difference(fb(X),lyx),Epsilon) )
& ( ~ $greatereq($difference(fb(X),lyx),0.0)
=> $less($uminus($difference(fb(X),lyx)),Epsilon) ) ) ) ) ) ).
%% (lxy = lyx)
tff(formula_5,conjecture,
lxy = lyx ).
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