TPTP Problem File: SEU032+1.p
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- Solve Problem
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% File : SEU032+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Set theory
% Problem : Functions and their basic properties, theorem 65
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [Byl90] Bylinski (1990), Functions and Their Basic Properties
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : funct_1__t65_funct_1 [Urb06]
% Status : Theorem
% Rating : 0.19 v8.2.0, 0.14 v8.1.0, 0.11 v7.5.0, 0.16 v7.4.0, 0.07 v7.3.0, 0.10 v7.1.0, 0.17 v7.0.0, 0.27 v6.4.0, 0.23 v6.3.0, 0.25 v6.2.0, 0.20 v6.1.0, 0.33 v6.0.0, 0.35 v5.5.0, 0.41 v5.4.0, 0.46 v5.3.0, 0.52 v5.2.0, 0.35 v5.1.0, 0.43 v5.0.0, 0.42 v4.1.0, 0.39 v4.0.0, 0.38 v3.7.0, 0.30 v3.5.0, 0.32 v3.4.0, 0.37 v3.3.0, 0.43 v3.2.0
% Syntax : Number of formulae : 44 ( 7 unt; 0 def)
% Number of atoms : 124 ( 10 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 94 ( 14 ~; 1 |; 49 &)
% ( 1 <=>; 29 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-2 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-2 aty)
% Number of variables : 60 ( 49 !; 11 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
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fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(dt_k2_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( relation(function_inverse(A))
& function(function_inverse(A)) ) ) ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(relation_composition(A,B)) ) ).
fof(dt_k6_relat_1,axiom,
! [A] : relation(identity_relation(A)) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc10_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(B,A))
& relation(relation_composition(B,A)) ) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& relation(B)
& function(B) )
=> ( relation(relation_composition(A,B))
& function(relation_composition(A,B)) ) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_funct_1,axiom,
! [A] :
( relation(identity_relation(A))
& function(identity_relation(A)) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(A)) ) ) ).
fof(fc9_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(A,B))
& relation(relation_composition(A,B)) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t55_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> ( relation_rng(A) = relation_dom(function_inverse(A))
& relation_dom(A) = relation_rng(function_inverse(A)) ) ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t61_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> ( relation_composition(A,function_inverse(A)) = identity_relation(relation_dom(A))
& relation_composition(function_inverse(A),A) = identity_relation(relation_rng(A)) ) ) ) ).
fof(t62_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> one_to_one(function_inverse(A)) ) ) ).
fof(t63_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( ( relation(B)
& function(B) )
=> ( ( one_to_one(A)
& relation_rng(A) = relation_dom(B)
& relation_composition(A,B) = identity_relation(relation_dom(A)) )
=> B = function_inverse(A) ) ) ) ).
fof(t65_funct_1,conjecture,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> function_inverse(function_inverse(A)) = A ) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
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