:: ABCMIZ_0 semantic presentation begin registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_lattice3 :::"complete"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; definitionlet "T" be ($#l1_orders_2 :::"RelStr"::: ) ; mode type of "T" is ($#m1_subset_1 :::"Element":::) "of" "T"; end; definitionlet "T" be ($#l1_orders_2 :::"RelStr"::: ) ; attr "T" is :::"Noetherian"::: means :: ABCMIZ_0:def 1 (Bool (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" "T") "is" ($#v1_rewrite1 :::"co-well_founded"::: ) ); end; :: deftheorem defines :::"Noetherian"::: ABCMIZ_0:def 1 : (Bool "for" (Set (Var "T")) "being" ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_abcmiz_0 :::"Noetherian"::: ) ) "iff" (Bool (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T"))) "is" ($#v1_rewrite1 :::"co-well_founded"::: ) ) ")" )); registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; redefine attr "T" is :::"Noetherian"::: means :: ABCMIZ_0:def 2 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "T" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "T" "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "not" (Bool (Set (Var "a")) ($#r2_orders_2 :::"<"::: ) (Set (Var "b")))) ")" ) ")" ))); end; :: deftheorem defines :::"Noetherian"::: ABCMIZ_0:def 2 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_abcmiz_0 :::"Noetherian"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "not" (Bool (Set (Var "a")) ($#r2_orders_2 :::"<"::: ) (Set (Var "b")))) ")" ) ")" ))) ")" )); definitionlet "T" be ($#l1_orders_2 :::"Poset":::); attr "T" is :::"Mizar-widening-like"::: means :: ABCMIZ_0:def 3 (Bool "(" (Bool "T" "is" ($#l1_orders_2 :::"sup-Semilattice":::)) & (Bool "T" "is" ($#v1_abcmiz_0 :::"Noetherian"::: ) ) ")" ); end; :: deftheorem defines :::"Mizar-widening-like"::: ABCMIZ_0:def 3 : (Bool "for" (Set (Var "T")) "being" ($#l1_orders_2 :::"Poset":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) ) "iff" (Bool "(" (Bool (Set (Var "T")) "is" ($#l1_orders_2 :::"sup-Semilattice":::)) & (Bool (Set (Var "T")) "is" ($#v1_abcmiz_0 :::"Noetherian"::: ) ) ")" ) ")" )); registration cluster ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) -> ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) -> ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) bbbadV3_YELLOW_0() ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "T" be ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" "T") -> ($#v1_rewrite1 :::"co-well_founded"::: ) ; end; theorem :: ABCMIZ_0:1 (Bool "for" (Set (Var "T")) "being" ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "I")) "," (Set (Var "T"))) & (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) ")" ))) ; begin definitionattr "c1" is :::"strict"::: ; struct :::"AdjectiveStr"::: -> ; aggr :::"AdjectiveStr":::(# :::"adjectives":::, :::"non-op"::: #) -> ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; sel :::"adjectives"::: "c1" -> ($#m1_hidden :::"set"::: ) ; sel :::"non-op"::: "c1" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "c1"); end; definitionlet "A" be ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; attr "A" is :::"void"::: means :: ABCMIZ_0:def 4 (Bool (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "A") "is" ($#v1_xboole_0 :::"empty"::: ) ); mode adjective of "A" is ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "A"); end; :: deftheorem defines :::"void"::: ABCMIZ_0:def 4 : (Bool "for" (Set (Var "A")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_abcmiz_0 :::"void"::: ) ) "iff" (Bool (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" )); theorem :: ABCMIZ_0:2 (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "st" (Bool (Bool (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A1"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A2")))) & (Bool (Set (Var "A1")) "is" ($#v4_abcmiz_0 :::"void"::: ) )) "holds" (Bool (Set (Var "A2")) "is" ($#v4_abcmiz_0 :::"void"::: ) )) ; definitionlet "A" be ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; let "a" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "A"))); func :::"non-"::: "a" -> ($#m1_subset_1 :::"adjective":::) "of" "A" equals :: ABCMIZ_0:def 5 (Set (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" "A") ($#k1_funct_1 :::"."::: ) "a"); end; :: deftheorem defines :::"non-"::: ABCMIZ_0:def 5 : (Bool "for" (Set (Var "A")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A"))) "holds" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A"))) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))))); theorem :: ABCMIZ_0:3 (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "st" (Bool (Bool (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A2"))) "#)" ))) "holds" (Bool "for" (Set (Var "a1")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "A1")) (Bool "for" (Set (Var "a2")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "A2")) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Var "a2")))) "holds" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a2"))))))) ; definitionlet "A" be ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; attr "A" is :::"involutive"::: means :: ABCMIZ_0:def 6 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "A" "holds" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a")))); attr "A" is :::"without_fixpoints"::: means :: ABCMIZ_0:def 7 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "A" "holds" (Bool (Bool "not" (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))))); end; :: deftheorem defines :::"involutive"::: ABCMIZ_0:def 6 : (Bool "for" (Set (Var "A")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v5_abcmiz_0 :::"involutive"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a")))) ")" )); :: deftheorem defines :::"without_fixpoints"::: ABCMIZ_0:def 7 : (Bool "for" (Set (Var "A")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v6_abcmiz_0 :::"without_fixpoints"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "A")) "holds" (Bool (Bool "not" (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ")" )); theorem :: ABCMIZ_0:4 (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"<>"::: ) (Set (Var "a2")))) "holds" (Bool "for" (Set (Var "A")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "st" (Bool (Bool (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A"))) ($#k1_funct_1 :::"."::: ) (Set (Var "a1"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A"))) ($#k1_funct_1 :::"."::: ) (Set (Var "a2"))) ($#r1_hidden :::"="::: ) (Set (Var "a1")))) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "A")) "is" ($#v4_abcmiz_0 :::"void"::: ) )) & (Bool (Set (Var "A")) "is" ($#v5_abcmiz_0 :::"involutive"::: ) ) & (Bool (Set (Var "A")) "is" ($#v6_abcmiz_0 :::"without_fixpoints"::: ) ) ")" ))) ; theorem :: ABCMIZ_0:5 (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "st" (Bool (Bool (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A2"))) "#)" )) & (Bool (Set (Var "A1")) "is" ($#v5_abcmiz_0 :::"involutive"::: ) )) "holds" (Bool (Set (Var "A2")) "is" ($#v5_abcmiz_0 :::"involutive"::: ) )) ; theorem :: ABCMIZ_0:6 (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) "st" (Bool (Bool (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_abcmiz_0 :::"AdjectiveStr"::: ) "(#" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "A2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "A2"))) "#)" )) & (Bool (Set (Var "A1")) "is" ($#v6_abcmiz_0 :::"without_fixpoints"::: ) )) "holds" (Bool (Set (Var "A2")) "is" ($#v6_abcmiz_0 :::"without_fixpoints"::: ) )) ; registration cluster ($#v3_abcmiz_0 :::"strict"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v5_abcmiz_0 :::"involutive"::: ) ($#v6_abcmiz_0 :::"without_fixpoints"::: ) for ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; end; registrationlet "A" be ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; cluster (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "A") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionattr "c1" is :::"strict"::: ; struct :::"TA-structure"::: -> ($#l1_orders_2 :::"RelStr"::: ) "," ($#l1_abcmiz_0 :::"AdjectiveStr"::: ) ; aggr :::"TA-structure":::(# :::"carrier":::, :::"adjectives":::, :::"InternalRel":::, :::"non-op":::, :::"adj-map"::: #) -> ($#l2_abcmiz_0 :::"TA-structure"::: ) ; sel :::"adj-map"::: "c1" -> ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1") "," (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "c1") ")" ); end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"set"::: ) ; let "r" be ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "X")); let "n" be ($#m1_subset_1 :::"UnOp":::) "of" (Set (Const "A")); let "a" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Const "A")) ")" ); cluster (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" "X" "," "A" "," "r" "," "n" "," "a" "#)" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "r" be ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "X")); let "n" be ($#m1_subset_1 :::"UnOp":::) "of" (Set (Const "A")); let "a" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Const "A")) ")" ); cluster (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" "X" "," "A" "," "r" "," "n" "," "a" "#)" ) -> ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ; end; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v5_abcmiz_0 :::"involutive"::: ) ($#v6_abcmiz_0 :::"without_fixpoints"::: ) ($#v7_abcmiz_0 :::"strict"::: ) for ($#l2_abcmiz_0 :::"TA-structure"::: ) ; end; definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); func :::"adjs"::: "t" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "T") equals :: ABCMIZ_0:def 8 (Set (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" "T") ($#k1_funct_1 :::"."::: ) "t"); end; :: deftheorem defines :::"adjs"::: ABCMIZ_0:def 8 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T"))) ($#k1_funct_1 :::"."::: ) (Set (Var "t")))))); theorem :: ABCMIZ_0:7 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T2"))) "#)" ))) "holds" (Bool "for" (Set (Var "t1")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T1")) (Bool "for" (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T2")) "st" (Bool (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Var "t2")))) "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2"))))))) ; definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; attr "T" is :::"consistent"::: means :: ABCMIZ_0:def 9 (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" "T" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))) "holds" (Bool "not" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))))); end; :: deftheorem defines :::"consistent"::: ABCMIZ_0:def 9 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v8_abcmiz_0 :::"consistent"::: ) ) "iff" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))) "holds" (Bool "not" (Bool (Set ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))))) ")" )); theorem :: ABCMIZ_0:8 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T2"))) "#)" )) & (Bool (Set (Var "T1")) "is" ($#v8_abcmiz_0 :::"consistent"::: ) )) "holds" (Bool (Set (Var "T2")) "is" ($#v8_abcmiz_0 :::"consistent"::: ) )) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; attr "T" is :::"adj-structured"::: means :: ABCMIZ_0:def 10 (Bool (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" "T") "is" ($#v20_waybel_0 :::"join-preserving"::: ) ($#m1_subset_1 :::"Function":::) "of" "T" "," (Set "(" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "T") ")" ) ($#k7_lattice3 :::"opp"::: ) ")" )); end; :: deftheorem defines :::"adj-structured"::: ABCMIZ_0:def 10 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v9_abcmiz_0 :::"adj-structured"::: ) ) "iff" (Bool (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T"))) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set "(" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) ")" ) ($#k7_lattice3 :::"opp"::: ) ")" )) ")" )); theorem :: ABCMIZ_0:9 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T2"))) "#)" )) & (Bool (Set (Var "T1")) "is" ($#v9_abcmiz_0 :::"adj-structured"::: ) )) "holds" (Bool (Set (Var "T2")) "is" ($#v9_abcmiz_0 :::"adj-structured"::: ) )) ; definitionlet "T" be ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; redefine attr "T" is :::"adj-structured"::: means :: ABCMIZ_0:def 11 (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set "(" (Set (Var "t1")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "t2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t1")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2")) ")" )))); end; :: deftheorem defines :::"adj-structured"::: ABCMIZ_0:def 11 : (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v9_abcmiz_0 :::"adj-structured"::: ) ) "iff" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set "(" (Set (Var "t1")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "t2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t1")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2")) ")" )))) ")" )); theorem :: ABCMIZ_0:10 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set (Var "T")) "is" ($#v9_abcmiz_0 :::"adj-structured"::: ) )) "holds" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "t1")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t2")))) "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t1")))))) ; definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "a" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); func :::"types"::: "a" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: ABCMIZ_0:def 12 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "t"))) & (Bool "a" ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t")))) ")" )) ")" )); end; :: deftheorem defines :::"types"::: ABCMIZ_0:def 12 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "t"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t")))) ")" )) ")" )) ")" )))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); func :::"types"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: ABCMIZ_0:def 13 (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "A")) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))))) ")" )); end; :: deftheorem defines :::"types"::: ABCMIZ_0:def 13 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_abcmiz_0 :::"types"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))))) ")" )) ")" )))); theorem :: ABCMIZ_0:11 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T2"))) "#)" ))) "holds" (Bool "for" (Set (Var "a1")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T1")) (Bool "for" (Set (Var "a2")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T2")) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Var "a2")))) "holds" (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a2"))))))) ; theorem :: ABCMIZ_0:12 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t")))) "}" ))) ; theorem :: ABCMIZ_0:13 (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t")))) "iff" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")))) ")" )))) ; theorem :: ABCMIZ_0:14 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t")))) "iff" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k4_abcmiz_0 :::"types"::: ) (Set (Var "A")))) ")" )))) ; theorem :: ABCMIZ_0:15 (Bool "for" (Set (Var "T")) "being" ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) : (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")))) "}" ))) ; theorem :: ABCMIZ_0:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) "holds" (Bool (Set ($#k4_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))))) ; definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; attr "T" is :::"adjs-typed"::: means :: ABCMIZ_0:def 14 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "holds" (Bool (Bool "not" (Set (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" )) "is" ($#v1_xboole_0 :::"empty"::: ) ))); end; :: deftheorem defines :::"adjs-typed"::: ABCMIZ_0:def 14 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v10_abcmiz_0 :::"adjs-typed"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Bool "not" (Set (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" )) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ")" )); theorem :: ABCMIZ_0:17 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) "st" (Bool (Bool (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T1"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g2_abcmiz_0 :::"TA-structure"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T2"))) "," (Set "the" ($#u3_abcmiz_0 :::"adj-map"::: ) "of" (Set (Var "T2"))) "#)" )) & (Bool (Set (Var "T1")) "is" ($#v10_abcmiz_0 :::"adjs-typed"::: ) )) "holds" (Bool (Set (Var "T2")) "is" ($#v10_abcmiz_0 :::"adjs-typed"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v8_struct_0 :::"finite"::: ) (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) bbbadV3_YELLOW_0() ($#v16_waybel_0 :::"connected"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v5_abcmiz_0 :::"involutive"::: ) ($#v6_abcmiz_0 :::"without_fixpoints"::: ) ($#v7_abcmiz_0 :::"strict"::: ) ($#v8_abcmiz_0 :::"consistent"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v10_abcmiz_0 :::"adjs-typed"::: ) for ($#l2_abcmiz_0 :::"TA-structure"::: ) ; end; theorem :: ABCMIZ_0:18 (Bool "for" (Set (Var "T")) "being" ($#v8_abcmiz_0 :::"consistent"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ))))) ; registrationlet "T" be ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "a" be ($#m1_subset_1 :::"adjective":::) "of" (Set (Const "T")); cluster (Set ($#k3_abcmiz_0 :::"types"::: ) "a") -> ($#v1_waybel_0 :::"directed"::: ) ($#v12_waybel_0 :::"lower"::: ) ; end; registrationlet "T" be ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); cluster (Set ($#k4_abcmiz_0 :::"types"::: ) "A") -> ($#v1_waybel_0 :::"directed"::: ) ($#v12_waybel_0 :::"lower"::: ) ; end; begin definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); let "a" be ($#m1_subset_1 :::"adjective":::) "of" (Set (Const "T")); pred "a" :::"is_applicable_to"::: "t" means :: ABCMIZ_0:def 15 (Bool "ex" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "t9")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) "a")) & (Bool (Set (Var "t9")) ($#r1_orders_2 :::"<="::: ) "t") ")" )); end; :: deftheorem defines :::"is_applicable_to"::: ABCMIZ_0:def 15 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "ex" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "t9")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")))) & (Bool (Set (Var "t9")) ($#r1_orders_2 :::"<="::: ) (Set (Var "t"))) ")" )) ")" )))); definitionlet "T" be ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); pred "A" :::"is_applicable_to"::: "t" means :: ABCMIZ_0:def 16 (Bool "ex" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool "(" (Bool "A" ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t9")))) & (Bool (Set (Var "t9")) ($#r1_orders_2 :::"<="::: ) "t") ")" )); end; :: deftheorem defines :::"is_applicable_to"::: ABCMIZ_0:def 16 : (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "ex" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t9")))) & (Bool (Set (Var "t9")) ($#r1_orders_2 :::"<="::: ) (Set (Var "t"))) ")" )) ")" )))); theorem :: ABCMIZ_0:19 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "t")) ")" )) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "T")))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); let "a" be ($#m1_subset_1 :::"adjective":::) "of" (Set (Const "T")); func "a" :::"ast"::: "t" -> ($#m1_subset_1 :::"type":::) "of" "T" equals :: ABCMIZ_0:def 17 (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) "a" ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) "t" ")" ) ")" )); end; :: deftheorem defines :::"ast"::: ABCMIZ_0:def 17 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "t")) ")" ) ")" )))))); theorem :: ABCMIZ_0:20 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:21 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set "(" (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_0:22 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))))))) ; theorem :: ABCMIZ_0:23 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "t9")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t9"))))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "t9")) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) ")" ))))) ; theorem :: ABCMIZ_0:24 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Var "t"))) ")" )))) ; theorem :: ABCMIZ_0:25 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "b")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))) "holds" (Bool "(" (Bool (Set (Var "b")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Set (Var "b")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "b")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k5_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" ))) ")" )))) ; theorem :: ABCMIZ_0:26 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set "(" ($#k4_abcmiz_0 :::"types"::: ) (Set (Var "A")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "t")) ")" )) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "T")))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); func "A" :::"ast"::: "t" -> ($#m1_subset_1 :::"type":::) "of" "T" equals :: ABCMIZ_0:def 18 (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k4_abcmiz_0 :::"types"::: ) "A" ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) "t" ")" ) ")" )); end; :: deftheorem defines :::"ast"::: ABCMIZ_0:def 18 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k4_abcmiz_0 :::"types"::: ) (Set (Var "A")) ")" ) ($#k3_finsub_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "t")) ")" ) ")" )))))); theorem :: ABCMIZ_0:27 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) ")" ) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Var "t"))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); func :::"apply"::: "(" "p" "," "t" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") means :: ABCMIZ_0:def 19 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) "p" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "t") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "p")) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set "p" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")))))) ")" ) ")" ); end; :: deftheorem defines :::"apply"::: ABCMIZ_0:def 19 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "p")) "," (Set (Var "t")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "t"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")))))) ")" ) ")" ) ")" ))))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); cluster (Set ($#k7_abcmiz_0 :::"apply"::: ) "(" "p" "," "t" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: ABCMIZ_0:28 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) ")" ) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "t")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: ABCMIZ_0:29 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k12_finseq_1 :::"*>"::: ) ) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "t")) "," (Set "(" (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" ) ($#k10_finseq_1 :::"*>"::: ) ))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "v" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); func "v" :::"ast"::: "t" -> ($#m1_subset_1 :::"type":::) "of" "T" equals :: ABCMIZ_0:def 20 (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" "v" "," "t" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) "v" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )); end; :: deftheorem defines :::"ast"::: ABCMIZ_0:def 20 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set (Set (Var "v")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "v")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))))); theorem :: ABCMIZ_0:30 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) ")" ) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Var "t"))))) ; theorem :: ABCMIZ_0:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))))) ; theorem :: ABCMIZ_0:32 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_rewrite1 :::"$^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))) ; theorem :: ABCMIZ_0:33 (Bool "for" (Set (Var "p")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_rewrite1 :::"$^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))))) ; theorem :: ABCMIZ_0:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set "(" (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2")) ")" ) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v1")) "," (Set (Var "t")) ")" ")" ) ($#k1_rewrite1 :::"$^"::: ) (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v2")) "," (Set "(" (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" ) ")" ")" )))))) ; theorem :: ABCMIZ_0:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "v1"))))) "holds" (Bool (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set "(" (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2")) ")" ) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v1")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))))) ; theorem :: ABCMIZ_0:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set "(" (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2")) ")" ) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "v1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))))) ; theorem :: ABCMIZ_0:37 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set (Set (Var "v2")) ($#k8_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2")) ")" ) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "v" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); pred "v" :::"is_applicable_to"::: "t" means :: ABCMIZ_0:def 21 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "v")) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set "v" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" "v" "," "t" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "s")))))); end; :: deftheorem defines :::"is_applicable_to"::: ABCMIZ_0:def 21 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "v")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "s")))))) ")" )))); theorem :: ABCMIZ_0:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T")))) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))))) ; theorem :: ABCMIZ_0:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k12_finseq_1 :::"*>"::: ) ) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) ")" )))) ; theorem :: ABCMIZ_0:40 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2"))) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "(" (Bool (Set (Var "v1")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "v2")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) ")" )))) ; theorem :: ABCMIZ_0:41 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "v")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i1")))) & (Bool (Set (Var "t2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i2"))))) "holds" (Bool (Set (Var "t2")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t1")))))))) ; theorem :: ABCMIZ_0:42 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t"))) ")" ))))) ; theorem :: ABCMIZ_0:43 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "v")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:44 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set "(" (Set (Var "v")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_0:45 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v"))))) "holds" (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))))))) ; theorem :: ABCMIZ_0:46 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v1")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v2"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v1"))))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v2")) "," (Set (Var "t")) ")" ")" )))) "holds" (Bool (Set (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "s"))))))) ; theorem :: ABCMIZ_0:47 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2"))) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "v2")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v1"))) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:48 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2"))) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set "(" (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2")) ")" ) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v2")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v1")) ")" ) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))))) ; theorem :: ABCMIZ_0:49 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:50 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set "(" (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_0:51 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_abcmiz_0 :::"types"::: ) (Set (Var "A"))))))) ; theorem :: ABCMIZ_0:52 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "t9")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "t9")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t9"))))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "t9")) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) ")" ))))) ; theorem :: ABCMIZ_0:53 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t"))))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Var "t"))) ")" )))) ; theorem :: ABCMIZ_0:54 (Bool "for" (Set (Var "T")) "being" ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "B")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:55 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_finsub_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "B")) ($#r2_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")))))))) ; theorem :: ABCMIZ_0:56 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v"))))) "holds" (Bool (Set (Set (Var "v")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")))))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) ; func :::"sub"::: "T" -> ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "T") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") means :: ABCMIZ_0:def 22 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" ) ")" )))); end; :: deftheorem defines :::"sub"::: ABCMIZ_0:def 22 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l2_abcmiz_0 :::"TA-structure"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_abcmiz_0 :::"sub"::: ) (Set (Var "T")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" ) ")" )))) ")" ))); definitionattr "c1" is :::"strict"::: ; struct :::"TAS-structure"::: -> ($#l2_abcmiz_0 :::"TA-structure"::: ) ; aggr :::"TAS-structure":::(# :::"carrier":::, :::"adjectives":::, :::"InternalRel":::, :::"non-op":::, :::"adj-map":::, :::"sub-map"::: #) -> ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; sel :::"sub-map"::: "c1" -> ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "c1") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); end; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v11_abcmiz_0 :::"strict"::: ) for ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; let "a" be ($#m1_subset_1 :::"adjective":::) "of" (Set (Const "T")); func :::"sub"::: "a" -> ($#m1_subset_1 :::"type":::) "of" "T" equals :: ABCMIZ_0:def 23 (Set (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" "T") ($#k3_funct_2 :::"."::: ) "a"); end; :: deftheorem defines :::"sub"::: ABCMIZ_0:def 23 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" (Set (Var "T"))) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; attr "T" is :::"non-absorbing"::: means :: ABCMIZ_0:def 24 (Bool (Set (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" "T") ($#k1_partfun1 :::"*"::: ) (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" "T")) ($#r2_funct_2 :::"="::: ) (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" "T")); attr "T" is :::"subjected"::: means :: ABCMIZ_0:def 25 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" )) ($#r2_lattice3 :::"is_<=_than"::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a")))) & "(" (Bool (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ))) ")" ")" )); end; :: deftheorem defines :::"non-absorbing"::: ABCMIZ_0:def 24 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v12_abcmiz_0 :::"non-absorbing"::: ) ) "iff" (Bool (Set (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" (Set (Var "T"))) ($#k1_partfun1 :::"*"::: ) (Set "the" ($#u2_abcmiz_0 :::"non-op"::: ) "of" (Set (Var "T")))) ($#r2_funct_2 :::"="::: ) (Set "the" ($#u4_abcmiz_0 :::"sub-map"::: ) "of" (Set (Var "T")))) ")" )); :: deftheorem defines :::"subjected"::: ABCMIZ_0:def 25 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v13_abcmiz_0 :::"subjected"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" )) ($#r2_lattice3 :::"is_<=_than"::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a")))) & "(" (Bool (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set (Var "a")) ")" ) ($#k1_finsub_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_0 :::"types"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ))) ")" ")" )) ")" )); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; redefine attr "T" is :::"non-absorbing"::: means :: ABCMIZ_0:def 26 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "holds" (Bool (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a"))))); end; :: deftheorem defines :::"non-absorbing"::: ABCMIZ_0:def 26 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v12_abcmiz_0 :::"non-absorbing"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set "(" ($#k1_abcmiz_0 :::"non-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a"))))) ")" )); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); let "a" be ($#m1_subset_1 :::"adjective":::) "of" (Set (Const "T")); pred "a" :::"is_properly_applicable_to"::: "t" means :: ABCMIZ_0:def 27 (Bool "(" (Bool "t" ($#r1_orders_2 :::"<="::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) "a")) & (Bool "a" ($#r1_abcmiz_0 :::"is_applicable_to"::: ) "t") ")" ); end; :: deftheorem defines :::"is_properly_applicable_to"::: ABCMIZ_0:def 27 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "(" (Bool (Set (Var "t")) ($#r1_orders_2 :::"<="::: ) (Set ($#k10_abcmiz_0 :::"sub"::: ) (Set (Var "a")))) & (Bool (Set (Var "a")) ($#r1_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t"))) ")" ) ")" )))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "v" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); pred "v" :::"is_properly_applicable_to"::: "t" means :: ABCMIZ_0:def 28 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "v")) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set "v" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" "v" "," "t" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "s")))))); end; :: deftheorem defines :::"is_properly_applicable_to"::: ABCMIZ_0:def 28 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "v")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "v")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "s")))))) ")" )))); theorem :: ABCMIZ_0:57 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Var "v")) ($#r3_abcmiz_0 :::"is_applicable_to"::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:58 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T")))) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))))) ; theorem :: ABCMIZ_0:59 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k12_finseq_1 :::"*>"::: ) ) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) ")" )))) ; theorem :: ABCMIZ_0:60 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2"))) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "(" (Bool (Set (Var "v1")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "v2")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) ")" )))) ; theorem :: ABCMIZ_0:61 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "v1")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Var "v2")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Set (Var "v1")) ($#k8_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))) "holds" (Bool (Set (Set (Var "v1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "v2"))) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Const "T"))); pred "A" :::"is_properly_applicable_to"::: "t" means :: ABCMIZ_0:def 29 (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "T") "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) "A") & (Bool (Set (Var "s")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) "t") ")" )); end; :: deftheorem defines :::"is_properly_applicable_to"::: ABCMIZ_0:def 29 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) "iff" (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "s")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) ")" )) ")" )))); theorem :: ABCMIZ_0:62 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) )))) ; theorem :: ABCMIZ_0:63 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T")))) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))))) ; scheme :: ABCMIZ_0:sch 1 MinimalFiniteSet{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool P1[(Set (Var "A"))]) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool P1[(Set (Var "B"))])) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ) ")" )) provided (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool P1[(Set (Var "A"))])) proof end; theorem :: ABCMIZ_0:64 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "B")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t")))) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "C")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "C")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "B"))) ")" ) ")" ))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; attr "T" is :::"commutative"::: means :: ABCMIZ_0:def 30 (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" "T" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "st" (Bool (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t1"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t1"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "t2")))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" "T") "st" (Bool "(" (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Set (Var "t1")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "t2")))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "t1")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "t2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "t2"))) ")" )))); end; :: deftheorem defines :::"commutative"::: ABCMIZ_0:def 30 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v14_abcmiz_0 :::"commutative"::: ) ) "iff" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t1"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t1"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "t2")))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Set (Var "t1")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "t2")))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set "(" (Set (Var "t1")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "t2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "t2"))) ")" )))) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v8_struct_0 :::"finite"::: ) (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) bbbadV3_YELLOW_0() ($#v16_waybel_0 :::"connected"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#v2_abcmiz_0 :::"Mizar-widening-like"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v5_abcmiz_0 :::"involutive"::: ) ($#v6_abcmiz_0 :::"without_fixpoints"::: ) ($#v8_abcmiz_0 :::"consistent"::: ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v10_abcmiz_0 :::"adjs-typed"::: ) ($#v11_abcmiz_0 :::"strict"::: ) ($#v12_abcmiz_0 :::"non-absorbing"::: ) ($#v13_abcmiz_0 :::"subjected"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) for ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; end; theorem :: ABCMIZ_0:65 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "s")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) ")" ))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; func "T" :::"@-->"::: -> ($#m1_subset_1 :::"Relation":::) "of" "T" means :: ABCMIZ_0:def 31 (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" "T" "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "t1")) "," (Set (Var "t2")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" "T" "st" (Bool "(" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2"))))) & (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t2"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Var "t1"))) ")" )) ")" )); end; :: deftheorem defines :::"@-->"::: ABCMIZ_0:def 31 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) )) "iff" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "t1")) "," (Set (Var "t2")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_0 :::"adjs"::: ) (Set (Var "t2"))))) & (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t2"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Var "t1"))) ")" )) ")" )) ")" ))); theorem :: ABCMIZ_0:66 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T"))))) ; scheme :: ABCMIZ_0:sch 2 RedInd{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ], F2() -> ($#m1_subset_1 :::"Relation":::) "of" (Set F1 "(" ")" ) } : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set F2 "(" ")" ) ($#r1_rewrite1 :::"reduces"::: ) (Set (Var "x")) "," (Set (Var "y")))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "y"))])) provided (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "y"))])) and (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "x"))])) and (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "x")) "," (Set (Var "y"))]) & (Bool P1[(Set (Var "y")) "," (Set (Var "z"))])) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "z"))])) proof end; theorem :: ABCMIZ_0:67 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) ($#r1_rewrite1 :::"reduces"::: ) (Set (Var "t1")) "," (Set (Var "t2")))) "holds" (Bool (Set (Var "t1")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t2"))))) ; theorem :: ABCMIZ_0:68 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) "is" ($#v2_relat_2 :::"irreflexive"::: ) )) ; theorem :: ABCMIZ_0:69 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) "is" ($#v3_rewrite1 :::"strongly-normalizing"::: ) )) ; theorem :: ABCMIZ_0:70 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "C")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "C")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "s")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "s")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set "(" (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "s")) "," (Set (Var "t")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ))))))) ; theorem :: ABCMIZ_0:71 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "C")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "C")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))))) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "s")) ($#r5_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k7_abcmiz_0 :::"apply"::: ) "(" (Set (Var "s")) "," (Set (Var "t")) ")" ")" )) "is" ($#m1_rewrite1 :::"RedSequence"::: ) "of" (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) )))))) ; theorem :: ABCMIZ_0:72 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) ($#r1_rewrite1 :::"reduces"::: ) (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t"))) "," (Set (Var "t")))))) ; theorem :: ABCMIZ_0:73 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r")) "being" ($#m1_rewrite1 :::"RedSequence"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "r")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")))))) ; theorem :: ABCMIZ_0:74 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "R")) ($#r1_rewrite1 :::"reduces"::: ) (Set (Var "x")) "," (Set (Var "y")))) "holds" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ABCMIZ_0:75 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "R")) "is" ($#v4_rewrite1 :::"with_UN_property"::: ) ) & (Bool (Set (Var "R")) "is" ($#v2_rewrite1 :::"weakly-normalizing"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_rewrite1 :::"nf"::: ) "(" (Set (Var "x")) "," (Set (Var "R")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: ABCMIZ_0:76 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) ($#r1_rewrite1 :::"reduces"::: ) (Set (Var "t1")) "," (Set (Var "t2")))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t2"))) & (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t2")))) ")" )))) ; theorem :: ABCMIZ_0:77 (Bool "for" (Set (Var "T")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) "is" ($#v8_rewrite1 :::"with_Church-Rosser_property"::: ) ) & (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) "is" ($#v4_rewrite1 :::"with_UN_property"::: ) ) ")" )) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) ; let "t" be ($#m1_subset_1 :::"type":::) "of" (Set (Const "T")); func :::"radix"::: "t" -> ($#m1_subset_1 :::"type":::) "of" "T" equals :: ABCMIZ_0:def 32 (Set ($#k2_rewrite1 :::"nf"::: ) "(" "t" "," (Set "(" "T" ($#k11_abcmiz_0 :::"@-->"::: ) ")" ) ")" ); end; :: deftheorem defines :::"radix"::: ABCMIZ_0:def 32 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rewrite1 :::"nf"::: ) "(" (Set (Var "t")) "," (Set "(" (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ")" ) ")" )))); theorem :: ABCMIZ_0:78 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "T")) ($#k11_abcmiz_0 :::"@-->"::: ) ) ($#r1_rewrite1 :::"reduces"::: ) (Set (Var "t")) "," (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:79 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "t")) ($#r3_orders_2 :::"<="::: ) (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t")))))) ; theorem :: ABCMIZ_0:80 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set (Var "t9")) where t9 "is" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_abcmiz_0 :::"adjectives"::: ) "of" (Set (Var "T"))) "st" (Bool "(" (Bool (Set (Var "A")) ($#r6_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t9"))) & (Bool (Set (Set (Var "A")) ($#k6_abcmiz_0 :::"ast"::: ) (Set (Var "t9"))) ($#r1_hidden :::"="::: ) (Set (Var "t"))) ")" )) "}" )) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "T"))) & (Bool (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "T")) ")" )) ")" )))) ; theorem :: ABCMIZ_0:81 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t1"))) & (Bool (Set (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t1"))) ($#r3_orders_2 :::"<="::: ) (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t2"))))) "holds" (Bool (Set (Var "t1")) ($#r3_orders_2 :::"<="::: ) (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t2"))))))) ; theorem :: ABCMIZ_0:82 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "t1")) ($#r3_orders_2 :::"<="::: ) (Set (Var "t2")))) "holds" (Bool (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t1"))) ($#r3_orders_2 :::"<="::: ) (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t2")))))) ; theorem :: ABCMIZ_0:83 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v1_abcmiz_0 :::"Noetherian"::: ) ($#~v4_abcmiz_0 "non" ($#v4_abcmiz_0 :::"void"::: ) ) ($#v9_abcmiz_0 :::"adj-structured"::: ) ($#v14_abcmiz_0 :::"commutative"::: ) ($#l3_abcmiz_0 :::"TAS-structure"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"type":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"adjective":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "a")) ($#r4_abcmiz_0 :::"is_properly_applicable_to"::: ) (Set (Var "t")))) "holds" (Bool (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set "(" (Set (Var "a")) ($#k5_abcmiz_0 :::"ast"::: ) (Set (Var "t")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_abcmiz_0 :::"radix"::: ) (Set (Var "t"))))))) ;