:: CONLAT_1 semantic presentation begin definitionlet "C" be ($#l5_struct_0 :::"2-sorted"::: ) ; attr "C" is :::"quasi-empty"::: means :: CONLAT_1:def 1 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ); end; :: deftheorem defines :::"quasi-empty"::: CONLAT_1:def 1 : (Bool "for" (Set (Var "C")) "being" ($#l5_struct_0 :::"2-sorted"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_conlat_1 :::"quasi-empty"::: ) ) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_struct_0 :::"strict"::: ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) for ($#l5_struct_0 :::"2-sorted"::: ) ; cluster ($#v10_struct_0 :::"strict"::: ) ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) for ($#l5_struct_0 :::"2-sorted"::: ) ; end; registration cluster ($#v2_struct_0 :::"empty"::: ) ($#v10_struct_0 :::"strict"::: ) ($#v11_struct_0 :::"void"::: ) ($#v1_conlat_1 :::"quasi-empty"::: ) for ($#l5_struct_0 :::"2-sorted"::: ) ; end; definitionattr "c1" is :::"strict"::: ; struct :::"ContextStr"::: -> ($#l5_struct_0 :::"2-sorted"::: ) ; aggr :::"ContextStr":::(# :::"carrier":::, :::"carrier'":::, :::"Information"::: #) -> ($#l1_conlat_1 :::"ContextStr"::: ) ; sel :::"Information"::: "c1" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1") "," (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "c1"); end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_conlat_1 :::"strict"::: ) for ($#l1_conlat_1 :::"ContextStr"::: ) ; cluster ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) ($#v2_conlat_1 :::"strict"::: ) for ($#l1_conlat_1 :::"ContextStr"::: ) ; end; definitionmode FormalContext is ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) ($#l1_conlat_1 :::"ContextStr"::: ) ; end; definitionlet "C" be ($#l5_struct_0 :::"2-sorted"::: ) ; mode Object of "C" is ($#m1_subset_1 :::"Element":::) "of" "C"; mode Attribute of "C" is ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C"); end; registrationlet "C" be ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) ($#l5_struct_0 :::"2-sorted"::: ) ; cluster (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "C" be ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) ($#l5_struct_0 :::"2-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C")); end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "o" be ($#m1_subset_1 :::"Object":::) "of" (Set (Const "C")); let "a" be ($#m1_subset_1 :::"Attribute":::) "of" (Set (Const "C")); pred "o" :::"is-connected-with"::: "a" means :: CONLAT_1:def 2 (Bool (Set ($#k4_tarski :::"["::: ) "o" "," "a" ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_conlat_1 :::"Information"::: ) "of" "C")); end; :: deftheorem defines :::"is-connected-with"::: CONLAT_1:def 2 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Attribute":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a"))) "iff" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "a")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_conlat_1 :::"Information"::: ) "of" (Set (Var "C")))) ")" )))); notationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "o" be ($#m1_subset_1 :::"Object":::) "of" (Set (Const "C")); let "a" be ($#m1_subset_1 :::"Attribute":::) "of" (Set (Const "C")); antonym "o" :::"is-not-connected-with"::: "a" for "o" :::"is-connected-with"::: "a"; end; begin definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"ObjectDerivation"::: "C" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") ")" ) means :: CONLAT_1:def 3 (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Attribute":::) "of" "C" : (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" "C" "st" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "O")))) "holds" (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a")))) "}" )); end; :: deftheorem defines :::"ObjectDerivation"::: CONLAT_1:def 3 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Attribute":::) "of" (Set (Var "C")) : (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "O")))) "holds" (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a")))) "}" )) ")" ))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"AttributeDerivation"::: "C" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ")" ) means :: CONLAT_1:def 4 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "o")) where o "is" ($#m1_subset_1 :::"Object":::) "of" "C" : (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Attribute":::) "of" "C" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a")))) "}" )); end; :: deftheorem defines :::"AttributeDerivation"::: CONLAT_1:def 4 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "o")) where o "is" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "C")) : (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Attribute":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a")))) "}" )) ")" ))); theorem :: CONLAT_1:1 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "o")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Attribute":::) "of" (Set (Var "C")) : (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a"))) "}" ))) ; theorem :: CONLAT_1:2 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Attribute":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "a")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set (Var "o")) where o "is" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "C")) : (Bool (Set (Var "o")) ($#r1_conlat_1 :::"is-connected-with"::: ) (Set (Var "a"))) "}" ))) ; theorem :: CONLAT_1:3 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O1")) "," (Set (Var "O2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "st" (Bool (Bool (Set (Var "O1")) ($#r1_tarski :::"c="::: ) (Set (Var "O2")))) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O2"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O1")))))) ; theorem :: CONLAT_1:4 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool (Bool (Set (Var "A1")) ($#r1_tarski :::"c="::: ) (Set (Var "A2")))) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A2"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A1")))))) ; theorem :: CONLAT_1:5 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Var "O")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ))))) ; theorem :: CONLAT_1:6 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" ))))) ; theorem :: CONLAT_1:7 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ")" ))))) ; theorem :: CONLAT_1:8 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: CONLAT_1:9 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "O")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")))) "iff" (Bool (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "O")) "," (Set (Var "A")) ($#k8_mcart_1 :::":]"::: ) ) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_conlat_1 :::"Information"::: ) "of" (Set (Var "C")))) ")" )))) ; theorem :: CONLAT_1:10 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")))) "iff" (Bool (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "O")) "," (Set (Var "A")) ($#k8_mcart_1 :::":]"::: ) ) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_conlat_1 :::"Information"::: ) "of" (Set (Var "C")))) ")" )))) ; theorem :: CONLAT_1:11 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "O")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")))) ")" )))) ; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"phi"::: "C" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ")" ) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") ")" ) equals :: CONLAT_1:def 5 (Set ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C"); end; :: deftheorem defines :::"phi"::: CONLAT_1:def 5 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k3_conlat_1 :::"phi"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C"))))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"psi"::: "C" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") ")" ) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ")" ) equals :: CONLAT_1:def 6 (Set ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C"); end; :: deftheorem defines :::"psi"::: CONLAT_1:def 6 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k4_conlat_1 :::"psi"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C"))))); definitionlet "P", "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "Con" be ($#m1_waybel_1 :::"Connection"::: ) "of" (Set (Const "P")) "," (Set (Const "R")); attr "Con" is :::"co-Galois"::: means :: CONLAT_1:def 7 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "P" "," "R"(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" "R" "," "P" "st" (Bool "(" (Bool "Con" ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel_1 :::"]"::: ) )) & (Bool (Set (Var "f")) "is" ($#v5_waybel_0 :::"antitone"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_waybel_0 :::"antitone"::: ) ) & (Bool "(" "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element":::) "of" "P" (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Element":::) "of" "R" "holds" (Bool "(" (Bool (Set (Var "p1")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1")) ")" ))) & (Bool (Set (Var "r1")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r1")) ")" ))) ")" )) ")" ) ")" ))); end; :: deftheorem defines :::"co-Galois"::: CONLAT_1:def 7 : (Bool "for" (Set (Var "P")) "," (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "Con")) "being" ($#m1_waybel_1 :::"Connection"::: ) "of" (Set (Var "P")) "," (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "Con")) "is" ($#v3_conlat_1 :::"co-Galois"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "P")) "," (Set (Var "R"))(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "P")) "st" (Bool "(" (Bool (Set (Var "Con")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel_1 :::"]"::: ) )) & (Bool (Set (Var "f")) "is" ($#v5_waybel_0 :::"antitone"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_waybel_0 :::"antitone"::: ) ) & (Bool "(" "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "P")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "p1")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1")) ")" ))) & (Bool (Set (Var "r1")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r1")) ")" ))) ")" )) ")" ) ")" ))) ")" ))); theorem :: CONLAT_1:12 (Bool "for" (Set (Var "P")) "," (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "Con")) "being" ($#m1_waybel_1 :::"Connection"::: ) "of" (Set (Var "P")) "," (Set (Var "R")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "P")) "," (Set (Var "R")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "P")) "st" (Bool (Bool (Set (Var "Con")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel_1 :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "Con")) "is" ($#v3_conlat_1 :::"co-Galois"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "P")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")))) "iff" (Bool (Set (Var "r")) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")))) ")" ))) ")" ))))) ; theorem :: CONLAT_1:13 (Bool "for" (Set (Var "P")) "," (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "Con")) "being" ($#m1_waybel_1 :::"Connection"::: ) "of" (Set (Var "P")) "," (Set (Var "R")) "st" (Bool (Bool (Set (Var "Con")) "is" ($#v3_conlat_1 :::"co-Galois"::: ) )) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "P")) "," (Set (Var "R")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "P")) "st" (Bool (Bool (Set (Var "Con")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel_1 :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "g")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ))) ")" ))))) ; theorem :: CONLAT_1:14 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set "(" ($#k3_conlat_1 :::"phi"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k4_conlat_1 :::"psi"::: ) (Set (Var "C")) ")" ) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_conlat_1 :::"co-Galois"::: ) )) ; theorem :: CONLAT_1:15 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O1")) "," (Set (Var "O2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "O1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "O2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O2")) ")" ))))) ; theorem :: CONLAT_1:16 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "A1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "A2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A2")) ")" ))))) ; theorem :: CONLAT_1:17 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))))) ; theorem :: CONLAT_1:18 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))))) ; begin definitionlet "C" be ($#l5_struct_0 :::"2-sorted"::: ) ; attr "c2" is :::"strict"::: ; struct :::"ConceptStr"::: "over" "C" -> ; aggr :::"ConceptStr":::(# :::"Extent":::, :::"Intent"::: #) -> ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; sel :::"Extent"::: "c2" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C"); sel :::"Intent"::: "c2" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C"); end; definitionlet "C" be ($#l5_struct_0 :::"2-sorted"::: ) ; let "CP" be ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Const "C")); attr "CP" is :::"empty"::: means :: CONLAT_1:def 8 (Bool "(" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP") "is" ($#v1_xboole_0 :::"empty"::: ) ) & (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" "CP") "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ); attr "CP" is :::"quasi-empty"::: means :: CONLAT_1:def 9 (Bool "(" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP") "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" "CP") "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ); end; :: deftheorem defines :::"empty"::: CONLAT_1:def 8 : (Bool "for" (Set (Var "C")) "being" ($#l5_struct_0 :::"2-sorted"::: ) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) "is" ($#v5_conlat_1 :::"empty"::: ) ) "iff" (Bool "(" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) & (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ) ")" ))); :: deftheorem defines :::"quasi-empty"::: CONLAT_1:def 9 : (Bool "for" (Set (Var "C")) "being" ($#l5_struct_0 :::"2-sorted"::: ) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) "is" ($#v6_conlat_1 :::"quasi-empty"::: ) ) "iff" (Bool "(" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ) ")" ))); registrationlet "C" be ($#~v1_conlat_1 "non" ($#v1_conlat_1 :::"quasi-empty"::: ) ) ($#l5_struct_0 :::"2-sorted"::: ) ; cluster ($#v4_conlat_1 :::"strict"::: ) ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; cluster ($#v4_conlat_1 :::"strict"::: ) ($#v6_conlat_1 :::"quasi-empty"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; registrationlet "C" be ($#v2_struct_0 :::"empty"::: ) ($#v11_struct_0 :::"void"::: ) ($#l5_struct_0 :::"2-sorted"::: ) ; cluster -> ($#v5_conlat_1 :::"empty"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; registrationlet "C" be ($#v1_conlat_1 :::"quasi-empty"::: ) ($#l5_struct_0 :::"2-sorted"::: ) ; cluster -> ($#v6_conlat_1 :::"quasi-empty"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP" be ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Const "C")); attr "CP" is :::"concept-like"::: means :: CONLAT_1:def 10 (Bool "(" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP")) ($#r1_hidden :::"="::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" "CP")) & (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" "CP")) ($#r1_hidden :::"="::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP")) ")" ); end; :: deftheorem defines :::"concept-like"::: CONLAT_1:def 10 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) "is" ($#v7_conlat_1 :::"concept-like"::: ) ) "iff" (Bool "(" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP")))) ($#r1_hidden :::"="::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP")))) & (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP")))) ($#r1_hidden :::"="::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP")))) ")" ) ")" ))); registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) ($#v7_conlat_1 :::"concept-like"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); mode FormalConcept of "C" is ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) ($#v7_conlat_1 :::"concept-like"::: ) ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster ($#v4_conlat_1 :::"strict"::: ) ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) ($#v7_conlat_1 :::"concept-like"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; theorem :: CONLAT_1:19 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ")" ) "," (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) & (Bool "(" "for" (Set (Var "O9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) (Bool "for" (Set (Var "A9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O9")) "," (Set (Var "A9")) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) & (Bool (Set (Var "O")) ($#r1_tarski :::"c="::: ) (Set (Var "O9")))) "holds" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "O9")))) ")" ) ")" ))) ; theorem :: CONLAT_1:20 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")))) "iff" (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "O"))) ")" ))) ; theorem :: CONLAT_1:21 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" ) "," (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" ) ")" ) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) & (Bool "(" "for" (Set (Var "O9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) (Bool "for" (Set (Var "A9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O9")) "," (Set (Var "A9")) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "A9")))) "holds" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A9")))) ")" ) ")" ))) ; theorem :: CONLAT_1:22 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "st" (Bool (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" ) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")))) "iff" (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ))) ; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP" be ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Const "C")); attr "CP" is :::"universal"::: means :: CONLAT_1:def 11 (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP") ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")); end; :: deftheorem defines :::"universal"::: CONLAT_1:def 11 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) "is" ($#v8_conlat_1 :::"universal"::: ) ) "iff" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) ")" ))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP" be ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Const "C")); attr "CP" is :::"co-universal"::: means :: CONLAT_1:def 12 (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" "CP") ($#r1_hidden :::"="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C")); end; :: deftheorem defines :::"co-universal"::: CONLAT_1:def 12 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) "is" ($#v9_conlat_1 :::"co-universal"::: ) ) "iff" (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C")))) ")" ))); registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster ($#v4_conlat_1 :::"strict"::: ) ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) ($#v7_conlat_1 :::"concept-like"::: ) ($#v8_conlat_1 :::"universal"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; cluster ($#v4_conlat_1 :::"strict"::: ) ($#~v5_conlat_1 "non" ($#v5_conlat_1 :::"empty"::: ) ) ($#v7_conlat_1 :::"concept-like"::: ) ($#v9_conlat_1 :::"co-universal"::: ) for ($#l2_conlat_1 :::"ConceptStr"::: ) "over" "C"; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"Concept-with-all-Objects"::: "C" -> ($#v4_conlat_1 :::"strict"::: ) ($#v8_conlat_1 :::"universal"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" "C" means :: CONLAT_1:def 13 (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ))) ")" ))); end; :: deftheorem defines :::"Concept-with-all-Objects"::: CONLAT_1:def 13 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#v8_conlat_1 :::"universal"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")))) "iff" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ))) ")" ))) ")" ))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"Concept-with-all-Attributes"::: "C" -> ($#v4_conlat_1 :::"strict"::: ) ($#v9_conlat_1 :::"co-universal"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" "C" means :: CONLAT_1:def 14 (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ))); end; :: deftheorem defines :::"Concept-with-all-Attributes"::: CONLAT_1:def 14 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#v9_conlat_1 :::"co-universal"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")))) "iff" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ))) ")" ))); theorem :: CONLAT_1:23 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool "(" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set "(" ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set "(" ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C")))) ")" )) ; theorem :: CONLAT_1:24 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" "(" (Bool (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "CP")) "is" ($#v9_conlat_1 :::"co-universal"::: ) ) ")" & "(" (Bool (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "CP")) "is" ($#v8_conlat_1 :::"universal"::: ) ) ")" ")" ))) ; theorem :: CONLAT_1:25 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" "(" (Bool (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "CP")) ($#r1_hidden :::"="::: ) (Set ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")))) ")" & "(" (Bool (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "CP")) ($#r1_hidden :::"="::: ) (Set ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")))) ")" ")" ))) ; theorem :: CONLAT_1:26 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v6_conlat_1 :::"quasi-empty"::: ) ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" "not" (Bool (Set (Var "CP")) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) "or" (Bool (Set (Var "CP")) "is" ($#v8_conlat_1 :::"universal"::: ) ) "or" (Bool (Set (Var "CP")) "is" ($#v9_conlat_1 :::"co-universal"::: ) ) ")" ))) ; theorem :: CONLAT_1:27 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v6_conlat_1 :::"quasi-empty"::: ) ($#l2_conlat_1 :::"ConceptStr"::: ) "over" (Set (Var "C")) "holds" (Bool "(" "not" (Bool (Set (Var "CP")) "is" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) "or" (Bool (Set (Var "CP")) ($#r1_hidden :::"="::: ) (Set ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")))) "or" (Bool (Set (Var "CP")) ($#r1_hidden :::"="::: ) (Set ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")))) ")" ))) ; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); mode :::"Set-of-FormalConcepts"::: "of" "C" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) means :: CONLAT_1:def 15 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Var "X")) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" "C")); end; :: deftheorem defines :::"Set-of-FormalConcepts"::: CONLAT_1:def 15 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_conlat_1 :::"Set-of-FormalConcepts"::: ) "of" (Set (Var "C"))) "iff" (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "b2")))) "holds" (Bool (Set (Var "X")) "is" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")))) ")" ))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "FCS" be ($#m1_conlat_1 :::"Set-of-FormalConcepts"::: ) "of" (Set (Const "C")); :: original: :::"Element"::: redefine mode :::"Element"::: "of" "FCS" -> ($#l2_conlat_1 :::"FormalConcept":::) "of" "C"; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP1", "CP2" be ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Const "C")); pred "CP1" :::"is-SubConcept-of"::: "CP2" means :: CONLAT_1:def 16 (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP1") ($#r1_tarski :::"c="::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" "CP2")); end; :: deftheorem defines :::"is-SubConcept-of"::: CONLAT_1:def 16 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP1")) ($#r2_conlat_1 :::"is-SubConcept-of"::: ) (Set (Var "CP2"))) "iff" (Bool (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP2")))) ")" ))); notationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP1", "CP2" be ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Const "C")); synonym "CP2" :::"is-SuperConcept-of"::: "CP1" for "CP1" :::"is-SubConcept-of"::: "CP2"; end; theorem :: CONLAT_1:28 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP1")) ($#r2_conlat_1 :::"is-SubConcept-of"::: ) (Set (Var "CP2"))) "iff" (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1")))) ")" ))) ; theorem :: CONLAT_1:29 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP1")) ($#r2_conlat_1 :::"is-SuperConcept-of"::: ) (Set (Var "CP2"))) "iff" (Bool (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2")))) ")" ))) ; theorem :: CONLAT_1:30 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "CP")) ($#r2_conlat_1 :::"is-SubConcept-of"::: ) (Set ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")))) & (Bool (Set ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C"))) ($#r2_conlat_1 :::"is-SubConcept-of"::: ) (Set (Var "CP"))) ")" ))) ; begin definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"B-carrier"::: "C" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) equals :: CONLAT_1:def 17 "{" (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "E")) "," (Set (Var "I")) "#)" ) where E "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C"), I "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") : (Bool "(" (Bool (Bool "not" (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "E")) "," (Set (Var "I")) "#)" ) "is" ($#v5_conlat_1 :::"empty"::: ) )) & (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "E"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))) & (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" ) "}" ; end; :: deftheorem defines :::"B-carrier"::: CONLAT_1:def 17 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k7_conlat_1 :::"B-carrier"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) "{" (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "E")) "," (Set (Var "I")) "#)" ) where E "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))), I "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) : (Bool "(" (Bool (Bool "not" (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "E")) "," (Set (Var "I")) "#)" ) "is" ($#v5_conlat_1 :::"empty"::: ) )) & (Bool (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "E"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))) & (Bool (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" ) "}" )); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); :: original: :::"B-carrier"::: redefine func :::"B-carrier"::: "C" -> ($#m1_conlat_1 :::"Set-of-FormalConcepts"::: ) "of" "C"; end; registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster (Set ($#k7_conlat_1 :::"B-carrier"::: ) "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CONLAT_1:31 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "CP")) ($#r2_hidden :::"in"::: ) (Set ($#k8_conlat_1 :::"B-carrier"::: ) (Set (Var "C")))) "iff" (Bool (Set (Var "CP")) "is" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C"))) ")" ))) ; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"B-meet"::: "C" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) "C" ")" ) means :: CONLAT_1:def 18 (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" "C" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "st" (Bool "(" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP1"))) ($#k9_subset_1 :::"/\"::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP2"))))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1"))) ($#k4_subset_1 :::"\/"::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2"))) ")" ) ")" ))) ")" )))); end; :: deftheorem defines :::"B-meet"::: CONLAT_1:def 18 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP1"))) ($#k9_subset_1 :::"/\"::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP2"))))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1"))) ($#k4_subset_1 :::"\/"::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2"))) ")" ) ")" ))) ")" )))) ")" ))); definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"B-join"::: "C" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) "C" ")" ) means :: CONLAT_1:def 19 (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" "C" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "st" (Bool "(" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) "C" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP1"))) ($#k4_subset_1 :::"\/"::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP2"))) ")" ) ")" ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1"))) ($#k9_subset_1 :::"/\"::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2"))))) ")" )))); end; :: deftheorem defines :::"B-join"::: CONLAT_1:def 19 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))(Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_conlat_1 :::"ConceptStr"::: ) "(#" (Set (Var "O")) "," (Set (Var "A")) "#)" )) & (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_conlat_1 :::"AttributeDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_conlat_1 :::"ObjectDerivation"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP1"))) ($#k4_subset_1 :::"\/"::: ) (Set "the" ($#u2_conlat_1 :::"Extent"::: ) "of" (Set (Var "CP2"))) ")" ) ")" ))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP1"))) ($#k9_subset_1 :::"/\"::: ) (Set "the" ($#u3_conlat_1 :::"Intent"::: ) "of" (Set (Var "CP2"))))) ")" )))) ")" ))); theorem :: CONLAT_1:32 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP2")) "," (Set (Var "CP1")) ")" )))) ; theorem :: CONLAT_1:33 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP2")) "," (Set (Var "CP1")) ")" )))) ; theorem :: CONLAT_1:34 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "," (Set (Var "CP3")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set "(" (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP2")) "," (Set (Var "CP3")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ")" ) "," (Set (Var "CP3")) ")" )))) ; theorem :: CONLAT_1:35 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "," (Set (Var "CP3")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set "(" (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP2")) "," (Set (Var "CP3")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ")" ) "," (Set (Var "CP3")) ")" )))) ; theorem :: CONLAT_1:36 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ")" ) "," (Set (Var "CP2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "CP2"))))) ; theorem :: CONLAT_1:37 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set "(" (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP1")) "," (Set (Var "CP2")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "CP1"))))) ; theorem :: CONLAT_1:38 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP")) "," (Set "(" ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "CP"))))) ; theorem :: CONLAT_1:39 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP")) "," (Set "(" ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_conlat_1 :::"Concept-with-all-Objects"::: ) (Set (Var "C")))))) ; theorem :: CONLAT_1:40 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP")) "," (Set "(" ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "CP"))))) ; theorem :: CONLAT_1:41 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "CP")) "," (Set "(" ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_conlat_1 :::"Concept-with-all-Attributes"::: ) (Set (Var "C")))))) ; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); func :::"ConceptLattice"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattices :::"strict"::: ) ($#l3_lattices :::"LattStr"::: ) equals :: CONLAT_1:def 20 (Set ($#g3_lattices :::"LattStr"::: ) "(#" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) "C" ")" ) "," (Set "(" ($#k10_conlat_1 :::"B-join"::: ) "C" ")" ) "," (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) "C" ")" ) "#)" ); end; :: deftheorem defines :::"ConceptLattice"::: CONLAT_1:def 20 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#g3_lattices :::"LattStr"::: ) "(#" (Set "(" ($#k8_conlat_1 :::"B-carrier"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k10_conlat_1 :::"B-join"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k9_conlat_1 :::"B-meet"::: ) (Set (Var "C")) ")" ) "#)" ))); theorem :: CONLAT_1:42 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Var "C"))) "is" ($#l3_lattices :::"Lattice":::))) ; registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster (Set ($#k11_conlat_1 :::"ConceptLattice"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattices :::"strict"::: ) ($#v10_lattices :::"Lattice-like"::: ) ; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Const "C")) ")" ); :: original: :::"Element"::: redefine mode :::"Element"::: "of" "S" -> ($#l2_conlat_1 :::"FormalConcept":::) "of" "C"; end; definitionlet "C" be ($#l1_conlat_1 :::"FormalContext":::); let "CP" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Const "C")) ")" ); func "CP" :::"@"::: -> ($#v4_conlat_1 :::"strict"::: ) ($#l2_conlat_1 :::"FormalConcept":::) "of" "C" equals :: CONLAT_1:def 21 "CP"; end; :: deftheorem defines :::"@"::: CONLAT_1:def 21 : (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Var "C")) ")" ) "holds" (Bool (Set (Set (Var "CP")) ($#k12_conlat_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "CP"))))); theorem :: CONLAT_1:43 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) (Bool "for" (Set (Var "CP1")) "," (Set (Var "CP2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set (Var "CP1")) ($#r3_lattices :::"[="::: ) (Set (Var "CP2"))) "iff" (Bool (Set (Set (Var "CP1")) ($#k12_conlat_1 :::"@"::: ) ) ($#r2_conlat_1 :::"is-SubConcept-of"::: ) (Set (Set (Var "CP2")) ($#k12_conlat_1 :::"@"::: ) )) ")" ))) ; theorem :: CONLAT_1:44 (Bool "for" (Set (Var "C")) "being" ($#l1_conlat_1 :::"FormalContext":::) "holds" (Bool (Set ($#k11_conlat_1 :::"ConceptLattice"::: ) (Set (Var "C"))) "is" ($#v4_lattice3 :::"complete"::: ) ($#l3_lattices :::"Lattice":::))) ; registrationlet "C" be ($#l1_conlat_1 :::"FormalContext":::); cluster (Set ($#k11_conlat_1 :::"ConceptLattice"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattices :::"strict"::: ) ($#v4_lattice3 :::"complete"::: ) ; end;