:: ABCMIZ_1 semantic presentation begin theorem :: ABCMIZ_1:1 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "f")))))) ; theorem :: ABCMIZ_1:2 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: ABCMIZ_1:3 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))))) ; theorem :: ABCMIZ_1:4 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "X")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) ; theorem :: ABCMIZ_1:5 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_dtconstr :::"Term":::) "of" (Set (Var "S")) "," (Set (Var "X")) "holds" (Bool (Bool "not" (Set (Var "t")) "is" ($#v1_xtuple_0 :::"pair"::: ) ))))) ; registrationlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); cluster -> ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" ))); end; theorem :: ABCMIZ_1:6 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "z")) ($#k1_tarski :::"}"::: ) ) ($#k3_finseq_2 :::"*"::: ) )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "z")) ($#k1_tarski :::"}"::: ) ) ($#k3_finseq_2 :::"*"::: ) )) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "y"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) ; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "A" be ($#l3_msualg_1 :::"MSAlgebra"::: ) "over" (Set (Const "S")); mode Subset of "A" is ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" "A") ")" ); mode FinSequence of "A" is ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" "A")); end; registrationlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "S")); cluster -> ($#v6_trees_3 :::"DTree-yielding"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" ))); end; theorem :: ABCMIZ_1:7 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "holds" (Bool "(" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "v")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set (Var "s")) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "X")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")))) ")" ))) "or" (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "st" (Bool "(" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ))) & (Bool (Set (Var "p")) "is" ($#v6_trees_3 :::"DTree-yielding"::: ) ) & (Bool (Set (Var "p")) "is" ($#m1_msaterm :::"ArgumentSeq"::: ) "of" (Set ($#k2_msaterm :::"Sym"::: ) "(" (Set (Var "o")) "," (Set "(" (Set (Var "X")) ($#k2_pboole :::"\/"::: ) (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k8_funcop_1 :::"-->"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ")" ))) ")" )))) ; definitionlet "A" be ($#m1_hidden :::"set"::: ) ; func :::"varcl"::: "A" -> ($#m1_hidden :::"set"::: ) means :: ABCMIZ_1:def 1 (Bool "(" (Bool "A" ($#r1_tarski :::"c="::: ) it) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) it) ")" ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "A" ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ) ")" ); projectivity (Bool "for" (Set (Var "b1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "b2")) ($#r1_tarski :::"c="::: ) (Set (Var "b1"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "b2")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "b1")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_tarski :::"c="::: ) (Set (Var "b1"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "b1")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "b1")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ) ")" ))) ; end; :: deftheorem defines :::"varcl"::: ABCMIZ_1:def 1 : (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "b2"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "b2"))) ")" ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "b2")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ) ")" ) ")" ))); theorem :: ABCMIZ_1:8 (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: ABCMIZ_1:9 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "B"))))) ; theorem :: ABCMIZ_1:10 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "a")) ")" ) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool verum) "}" ))) ; scheme :: ABCMIZ_1:sch 1 Sch14{ F1() -> ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set F2 "(" (Set (Var "z")) ")" ) where z "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) : (Bool P1[(Set (Var "z"))]) "}" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set F2 "(" (Set (Var "z")) ")" ) ")" ) where z "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) : (Bool P1[(Set (Var "z"))]) "}" )) proof end; theorem :: ABCMIZ_1:11 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "X")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "Y")) ")" )))) ; theorem :: ABCMIZ_1:12 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" )) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "A"))))) ; theorem :: ABCMIZ_1:13 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "X")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "Y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "X")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "Y")) ")" )))) ; registrationlet "A" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_abcmiz_1 :::"varcl"::: ) "A") -> ($#v1_xboole_0 :::"empty"::: ) ; end; definitionfunc :::"Vars"::: -> ($#m1_hidden :::"set"::: ) means :: ABCMIZ_1:def 2 (Bool "ex" (Set (Var "V")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "V")))) & (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ), j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ) ")" ) ")" )); end; :: deftheorem defines :::"Vars"::: ABCMIZ_1:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) "iff" (Bool "ex" (Set (Var "V")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "V")))) & (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ), j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ) ")" ) ")" )) ")" )); theorem :: ABCMIZ_1:14 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ), j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ) ")" )) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))))) ; theorem :: ABCMIZ_1:15 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ), j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ) ")" )) "holds" (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; theorem :: ABCMIZ_1:16 (Bool "{" (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ($#r1_tarski :::"c="::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ; theorem :: ABCMIZ_1:17 (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )))) ; theorem :: ABCMIZ_1:18 (Bool (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ), j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ) ; theorem :: ABCMIZ_1:19 (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ; theorem :: ABCMIZ_1:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k6_classes1 :::"the_rank_of"::: ) (Set (Var "X"))) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool (Set (Var "X")) "is" ($#v1_finset_1 :::"finite"::: ) )) ; theorem :: ABCMIZ_1:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k6_classes1 :::"the_rank_of"::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_classes1 :::"the_rank_of"::: ) (Set (Var "X"))))) ; theorem :: ABCMIZ_1:22 (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_classes1 :::"Rank"::: ) (Set ($#k4_ordinal1 :::"omega"::: ) ) ")" ) "holds" (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set ($#k4_classes1 :::"Rank"::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )))) ; theorem :: ABCMIZ_1:23 (Bool (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_classes1 :::"Rank"::: ) (Set ($#k4_ordinal1 :::"omega"::: ) ))) ; theorem :: ABCMIZ_1:24 (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A"))) "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ))) ; registration cluster (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionmode variable is ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ); end; registrationlet "x" be ($#m1_subset_1 :::"variable":::); cluster (Set ($#k1_xtuple_0 :::"vars"::: ) "x") -> ($#v1_finset_1 :::"finite"::: ) ; end; notationlet "x" be ($#m1_subset_1 :::"variable":::); synonym :::"vars"::: "x" for "x" :::"`1"::: ; end; definitionlet "x" be ($#m1_subset_1 :::"variable":::); :: original: :::"vars"::: redefine func :::"vars"::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ); end; theorem :: ABCMIZ_1:25 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) ))) ; theorem :: ABCMIZ_1:26 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: ABCMIZ_1:27 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: ABCMIZ_1:28 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )))) ; begin notationlet "R" be ($#m1_hidden :::"Relation":::); let "A" be ($#m1_hidden :::"set"::: ) ; synonym "R" :::"dom"::: "A" for "R" :::"|"::: "A"; end; definitionfunc :::"QuasiLoci"::: -> ($#m1_finseq_2 :::"FinSequenceSet"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) means :: ABCMIZ_1:def 3 (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_relat_1 :::"dom"::: ) (Set (Var "i")) ")" ))) ")" ) ")" ) ")" )); end; :: deftheorem defines :::"QuasiLoci"::: ABCMIZ_1:def 3 : (Bool "for" (Set (Var "b1")) "being" ($#m1_finseq_2 :::"FinSequenceSet"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) "iff" (Bool "(" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_relat_1 :::"dom"::: ) (Set (Var "i")) ")" ))) ")" ) ")" ) ")" )) ")" )); theorem :: ABCMIZ_1:29 (Bool (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ($#r2_hidden :::"in"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) )) ; registration cluster (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionmode quasi-loci is ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ); end; registration cluster -> ($#v2_funct_1 :::"one-to-one"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ); end; theorem :: ABCMIZ_1:30 (Bool "for" (Set (Var "l")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#m2_finseq_2 :::"quasi-loci":::)) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "l")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "l")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x"))))) "holds" (Bool "ex" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "l")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "l")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" ))))) ")" )) ; theorem :: ABCMIZ_1:31 (Bool "for" (Set (Var "l")) "being" ($#m2_finseq_2 :::"quasi-loci":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" (Bool (Set (Set (Var "l")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "x")) ($#k3_pre_poly :::"*>"::: ) )) "is" ($#m2_finseq_2 :::"quasi-loci":::)) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "l"))))) & (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "l")))) ")" ) ")" ))) ; theorem :: ABCMIZ_1:32 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) "is" ($#m2_finseq_2 :::"quasi-loci":::))) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#m2_finseq_2 :::"quasi-loci":::)) & (Bool (Set (Var "q")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ")" )) ; theorem :: ABCMIZ_1:33 (Bool "for" (Set (Var "l")) "being" ($#m2_finseq_2 :::"quasi-loci":::) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "l")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "l"))))) ; theorem :: ABCMIZ_1:34 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" (Bool (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "x")) ($#k3_pre_poly :::"*>"::: ) ) "is" ($#m2_finseq_2 :::"quasi-loci":::)) "iff" (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; theorem :: ABCMIZ_1:35 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" (Bool (Set ($#k4_pre_poly :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_pre_poly :::"*>"::: ) ) "is" ($#m2_finseq_2 :::"quasi-loci":::)) "iff" (Bool "(" (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "y"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ")" ) ")" )) ; theorem :: ABCMIZ_1:36 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) "is" ($#m2_finseq_2 :::"quasi-loci":::)) "iff" (Bool "(" (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "y"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "z"))) ($#r1_tarski :::"c="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k7_domain_1 :::"}"::: ) )) ")" ) ")" )) ; definitionlet "l" be ($#m2_finseq_2 :::"quasi-loci":::); :: original: :::"""::: redefine func "l" :::"""::: -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ); end; begin definitionfunc :::"a_Type"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 4 (Set ($#k6_numbers :::"0"::: ) ); func :::"an_Adj"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 5 (Num 1); func :::"a_Term"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 6 (Num 2); func :::"*"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 7 (Set ($#k6_numbers :::"0"::: ) ); func :::"non_op"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 8 (Num 1); end; :: deftheorem defines :::"a_Type"::: ABCMIZ_1:def 4 : (Bool (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); :: deftheorem defines :::"an_Adj"::: ABCMIZ_1:def 5 : (Bool (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)); :: deftheorem defines :::"a_Term"::: ABCMIZ_1:def 6 : (Bool (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#r1_hidden :::"="::: ) (Num 2)); :: deftheorem defines :::"*"::: ABCMIZ_1:def 7 : (Bool (Set ($#k9_abcmiz_1 :::"*"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); :: deftheorem defines :::"non_op"::: ABCMIZ_1:def 8 : (Bool (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)); definitionlet "C" be ($#l1_msualg_1 :::"Signature":::); attr "C" is :::"constructor"::: means :: ABCMIZ_1:def 9 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) "," (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) "," (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C")) & (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" "C") ($#k1_funct_1 :::"."::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) "," (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" "C") ($#k1_funct_1 :::"."::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" "C") ($#k1_funct_1 :::"."::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) & (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" "C") ($#k1_funct_1 :::"."::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) & (Bool "(" "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C") "st" (Bool (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) & (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) ))) "holds" (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" "C") ($#k1_funct_1 :::"."::: ) (Set (Var "o"))) ($#r2_hidden :::"in"::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k3_finseq_2 :::"*"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"constructor"::: ABCMIZ_1:def 9 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"Signature":::) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_abcmiz_1 :::"constructor"::: ) ) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) "," (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) "," (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C")))) & (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" (Set (Var "C"))) ($#k1_funct_1 :::"."::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) "," (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" (Set (Var "C"))) ($#k1_funct_1 :::"."::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" (Set (Var "C"))) ($#k1_funct_1 :::"."::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) & (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" (Set (Var "C"))) ($#k1_funct_1 :::"."::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) & (Bool "(" "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "C"))) "st" (Bool (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) & (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) ))) "holds" (Bool (Set (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" (Set (Var "C"))) ($#k1_funct_1 :::"."::: ) (Set (Var "o"))) ($#r2_hidden :::"in"::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k3_finseq_2 :::"*"::: ) )) ")" ) ")" ) ")" )); registration cluster bbbadV1_INSTALG1() ($#v1_abcmiz_1 :::"constructor"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) for ($#l1_msualg_1 :::"ManySortedSign"::: ) ; end; definitionfunc :::"MinConstrSign"::: -> ($#v1_msualg_1 :::"strict"::: ) ($#l1_msualg_1 :::"Signature":::) means :: ABCMIZ_1:def 10 (Bool "(" (Bool it "is" ($#v1_abcmiz_1 :::"constructor"::: ) ) & (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) )) ")" ); end; :: deftheorem defines :::"MinConstrSign"::: ABCMIZ_1:def 10 : (Bool "for" (Set (Var "b1")) "being" ($#v1_msualg_1 :::"strict"::: ) ($#l1_msualg_1 :::"Signature":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k11_abcmiz_1 :::"MinConstrSign"::: ) )) "iff" (Bool "(" (Bool (Set (Var "b1")) "is" ($#v1_abcmiz_1 :::"constructor"::: ) ) & (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "b1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) )) ")" ) ")" )); registration cluster (Set ($#k11_abcmiz_1 :::"MinConstrSign"::: ) ) -> ($#v1_msualg_1 :::"strict"::: ) ($#v1_abcmiz_1 :::"constructor"::: ) ; end; registration cluster ($#v1_msualg_1 :::"strict"::: ) bbbadV1_INSTALG1() ($#v1_abcmiz_1 :::"constructor"::: ) for ($#l1_msualg_1 :::"ManySortedSign"::: ) ; end; definitionmode ConstructorSignature is ($#v1_abcmiz_1 :::"constructor"::: ) ($#l1_msualg_1 :::"Signature":::); end; definitionlet "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); let "o" be ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "C")); attr "o" is :::"constructor"::: means :: ABCMIZ_1:def 11 (Bool "(" (Bool "o" ($#r1_hidden :::"<>"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) & (Bool "o" ($#r1_hidden :::"<>"::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ")" ); end; :: deftheorem defines :::"constructor"::: ABCMIZ_1:def 11 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "o")) "is" ($#v2_abcmiz_1 :::"constructor"::: ) ) "iff" (Bool "(" (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) )) & (Bool (Set (Var "o")) ($#r1_hidden :::"<>"::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) )) ")" ) ")" ))); theorem :: ABCMIZ_1:37 (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "o")) "is" ($#v2_abcmiz_1 :::"constructor"::: ) )) "holds" (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ")" ) ($#k2_finseq_2 :::"|->"::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ))))) ; definitionlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); attr "C" is :::"initialized"::: means :: ABCMIZ_1:def 12 (Bool "ex" (Set (Var "m")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" "C" "st" (Bool "(" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )); end; :: deftheorem defines :::"initialized"::: ABCMIZ_1:def 12 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v3_abcmiz_1 :::"initialized"::: ) ) "iff" (Bool "ex" (Set (Var "m")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ")" )); definitionlet "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"a_Type"::: "C" -> ($#m1_subset_1 :::"SortSymbol":::) "of" "C" equals :: ABCMIZ_1:def 13 (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ); func :::"an_Adj"::: "C" -> ($#m1_subset_1 :::"SortSymbol":::) "of" "C" equals :: ABCMIZ_1:def 14 (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ); func :::"a_Term"::: "C" -> ($#m1_subset_1 :::"SortSymbol":::) "of" "C" equals :: ABCMIZ_1:def 15 (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ); func :::"non_op"::: "C" -> ($#m1_subset_1 :::"OperSymbol":::) "of" "C" equals :: ABCMIZ_1:def 16 (Set ($#k10_abcmiz_1 :::"non_op"::: ) ); func :::"ast"::: "C" -> ($#m1_subset_1 :::"OperSymbol":::) "of" "C" equals :: ABCMIZ_1:def 17 (Set ($#k9_abcmiz_1 :::"*"::: ) ); end; :: deftheorem defines :::"a_Type"::: ABCMIZ_1:def 13 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ))); :: deftheorem defines :::"an_Adj"::: ABCMIZ_1:def 14 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ))); :: deftheorem defines :::"a_Term"::: ABCMIZ_1:def 15 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k14_abcmiz_1 :::"a_Term"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ))); :: deftheorem defines :::"non_op"::: ABCMIZ_1:def 16 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) ))); :: deftheorem defines :::"ast"::: ABCMIZ_1:def 17 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ))); theorem :: ABCMIZ_1:38 (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set "(" ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")) ")" ) ($#k3_pre_poly :::"*>"::: ) )) & (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_pre_poly :::"<*"::: ) (Set "(" ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")) ")" ) ($#k4_pre_poly :::"*>"::: ) )) & (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")))) ")" )) ; definitionfunc :::"Modes"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 18 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); func :::"Attrs"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 19 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); func :::"Funcs"::: -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 20 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); end; :: deftheorem defines :::"Modes"::: ABCMIZ_1:def 18 : (Bool (Set ($#k17_abcmiz_1 :::"Modes"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); :: deftheorem defines :::"Attrs"::: ABCMIZ_1:def 19 : (Bool (Set ($#k18_abcmiz_1 :::"Attrs"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); :: deftheorem defines :::"Funcs"::: ABCMIZ_1:def 20 : (Bool (Set ($#k19_abcmiz_1 :::"Funcs"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); registration cluster (Set ($#k17_abcmiz_1 :::"Modes"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k18_abcmiz_1 :::"Attrs"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k19_abcmiz_1 :::"Funcs"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionfunc :::"Constructors"::: -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 21 (Set (Set "(" (Set ($#k17_abcmiz_1 :::"Modes"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k18_abcmiz_1 :::"Attrs"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k19_abcmiz_1 :::"Funcs"::: ) )); end; :: deftheorem defines :::"Constructors"::: ABCMIZ_1:def 21 : (Bool (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k17_abcmiz_1 :::"Modes"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k18_abcmiz_1 :::"Attrs"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k19_abcmiz_1 :::"Funcs"::: ) ))); theorem :: ABCMIZ_1:39 (Bool (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#r1_subset_1 :::"misses"::: ) (Set ($#k20_abcmiz_1 :::"Constructors"::: ) )) ; definitionlet "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); :: original: :::"vars"::: redefine func "x" :::"`1"::: -> ($#m2_finseq_2 :::"quasi-loci":::); :: original: :::"the_base_of"::: redefine func "x" :::"`2"::: -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; notationlet "c" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ); synonym :::"kind_of"::: "c" for "c" :::"`1"::: ; end; definitionlet "c" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ); :: original: :::"vars"::: redefine func :::"kind_of"::: "c" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ) "," (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ) "," (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k1_enumset1 :::"}"::: ) ); :: original: :::"the_base_of"::: redefine func "c" :::"`2"::: -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_abcmiz_1 :::"QuasiLoci"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); end; definitionlet "c" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ); func :::"loci_of"::: "c" -> ($#m2_finseq_2 :::"quasi-loci":::) equals :: ABCMIZ_1:def 22 (Set (Set "(" "c" ($#k24_abcmiz_1 :::"`2"::: ) ")" ) ($#k21_abcmiz_1 :::"`1"::: ) ); func :::"index_of"::: "c" -> ($#m1_hidden :::"Nat":::) equals :: ABCMIZ_1:def 23 (Set (Set "(" "c" ($#k24_abcmiz_1 :::"`2"::: ) ")" ) ($#k22_abcmiz_1 :::"`2"::: ) ); end; :: deftheorem defines :::"loci_of"::: ABCMIZ_1:def 22 : (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ) "holds" (Bool (Set ($#k25_abcmiz_1 :::"loci_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "c")) ($#k24_abcmiz_1 :::"`2"::: ) ")" ) ($#k21_abcmiz_1 :::"`1"::: ) ))); :: deftheorem defines :::"index_of"::: ABCMIZ_1:def 23 : (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ) "holds" (Bool (Set ($#k26_abcmiz_1 :::"index_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "c")) ($#k24_abcmiz_1 :::"`2"::: ) ")" ) ($#k22_abcmiz_1 :::"`2"::: ) ))); theorem :: ABCMIZ_1:40 (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) ))) "implies" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k17_abcmiz_1 :::"Modes"::: ) )) ")" & "(" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k17_abcmiz_1 :::"Modes"::: ) ))) "implies" (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) ")" & "(" (Bool (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) ))) "implies" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k18_abcmiz_1 :::"Attrs"::: ) )) ")" & "(" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k18_abcmiz_1 :::"Attrs"::: ) ))) "implies" (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) ")" & "(" (Bool (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ))) "implies" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k19_abcmiz_1 :::"Funcs"::: ) )) ")" & "(" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k19_abcmiz_1 :::"Funcs"::: ) ))) "implies" (Bool (Set ($#k23_abcmiz_1 :::"kind_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) )) ")" ")" )) ; definitionfunc :::"MaxConstrSign"::: -> ($#v1_msualg_1 :::"strict"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) means :: ABCMIZ_1:def 24 (Bool "(" (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ))) & (Bool "(" "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" it "st" (Bool (Bool (Set (Var "o")) "is" ($#v2_abcmiz_1 :::"constructor"::: ) )) "holds" (Bool "(" (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" it) ($#k3_funct_2 :::"."::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "o")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" it) ($#k3_funct_2 :::"."::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "o")) ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k1_xtuple_0 :::"`1"::: ) ")" ))) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"MaxConstrSign"::: ABCMIZ_1:def 24 : (Bool "for" (Set (Var "b1")) "being" ($#v1_msualg_1 :::"strict"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k27_abcmiz_1 :::"MaxConstrSign"::: ) )) "iff" (Bool "(" (Bool (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "b1"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_tarski :::"{"::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k20_abcmiz_1 :::"Constructors"::: ) ))) & (Bool "(" "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "b1")) "st" (Bool (Bool (Set (Var "o")) "is" ($#v2_abcmiz_1 :::"constructor"::: ) )) "holds" (Bool "(" (Bool (Set (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" (Set (Var "b1"))) ($#k3_funct_2 :::"."::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "o")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set "the" ($#u1_msualg_1 :::"Arity"::: ) "of" (Set (Var "b1"))) ($#k3_funct_2 :::"."::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "o")) ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k1_xtuple_0 :::"`1"::: ) ")" ))) ")" ) ")" ) ")" ) ")" )); registration cluster (Set ($#k11_abcmiz_1 :::"MinConstrSign"::: ) ) -> ($#v1_msualg_1 :::"strict"::: ) ($#~v3_abcmiz_1 "non" ($#v3_abcmiz_1 :::"initialized"::: ) ) ; cluster (Set ($#k27_abcmiz_1 :::"MaxConstrSign"::: ) ) -> ($#v1_msualg_1 :::"strict"::: ) ($#v3_abcmiz_1 :::"initialized"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) bbbadV14_STRUCT_0() ($#v1_msualg_1 :::"strict"::: ) bbbadV1_INSTALG1() ($#v1_abcmiz_1 :::"constructor"::: ) ($#v3_abcmiz_1 :::"initialized"::: ) for ($#l1_msualg_1 :::"ManySortedSign"::: ) ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#v2_abcmiz_1 :::"constructor"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "C"); end; begin definitionlet "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"MSVars"::: "C" -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") means :: ABCMIZ_1:def 25 (Bool "(" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ")" ); end; :: deftheorem defines :::"MSVars"::: ABCMIZ_1:def 25 : (Bool "for" (Set (Var "C")) "being" ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set ($#k8_abcmiz_1 :::"a_Term"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) ")" ) ")" ))); registrationlet "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k28_abcmiz_1 :::"MSVars"::: ) "C") -> bbbadV3_RELAT_1() ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ) -> ($#v4_msualg_1 :::"non-empty"::: ) ; end; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Const "S")) "," (Set (Const "X")) ")" ")" ); attr "t" is :::"ground"::: means :: ABCMIZ_1:def 26 (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" "S" ($#k2_msafree3 :::"variables_in"::: ) "t" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); attr "t" is :::"compound"::: means :: ABCMIZ_1:def 27 (Bool (Set "t" ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r2_hidden :::"in"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" "S") "," (Set ($#k1_tarski :::"{"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; :: deftheorem defines :::"ground"::: ABCMIZ_1:def 26 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "t")) "is" ($#v4_abcmiz_1 :::"ground"::: ) ) "iff" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )))); :: deftheorem defines :::"compound"::: ABCMIZ_1:def 27 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "t")) "is" ($#v5_abcmiz_1 :::"compound"::: ) ) "iff" (Bool (Set (Set (Var "t")) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r2_hidden :::"in"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_tarski :::"{"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" )))); definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); mode expression of "C" is ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" ); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "s" be ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "C")); mode :::"expression"::: "of" "C" "," "s" -> ($#m1_subset_1 :::"expression":::) "of" "C" means :: ABCMIZ_1:def 28 (Bool it ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) "s")); end; :: deftheorem defines :::"expression"::: ABCMIZ_1:def 28 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s"))) "iff" (Bool (Set (Var "b3")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")))) ")" )))); theorem :: ABCMIZ_1:41 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "z")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s"))) "iff" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")))) ")" )))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "c" be ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "C")); assume (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "c")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ; func "c" :::"term"::: -> ($#m1_subset_1 :::"expression":::) "of" "C" equals :: ABCMIZ_1:def 29 (Set (Set ($#k4_tarski :::"["::: ) "c" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"term"::: ABCMIZ_1:def 29 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "c")) ($#k29_abcmiz_1 :::"term"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))); theorem :: ABCMIZ_1:42 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "a")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s"))) ")" ))) "holds" (Bool (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "a")) ($#k3_pre_poly :::"*>"::: ) )) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "o"))))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "o" be ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "C")); assume (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ; let "e" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); assume (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "C")) "st" (Bool "(" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Const "e")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set (Var "s"))) ")" )) ; func "o" :::"term"::: "e" -> ($#m1_subset_1 :::"expression":::) "of" "C" equals :: ABCMIZ_1:def 30 (Set (Set ($#k4_tarski :::"["::: ) "o" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k3_pre_poly :::"<*"::: ) "e" ($#k3_pre_poly :::"*>"::: ) )); end; :: deftheorem defines :::"term"::: ABCMIZ_1:def 30 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "e")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s"))) ")" ))) "holds" (Bool (Set (Set (Var "o")) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "e")) ($#k3_pre_poly :::"*>"::: ) )))))); theorem :: ABCMIZ_1:43 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) & (Bool (Set (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "a")) ($#k3_pre_poly :::"*>"::: ) ))) ")" ))) ; theorem :: ABCMIZ_1:44 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "b"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b"))))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); cluster (Set (Set "(" ($#k15_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) "a") -> ($#v5_abcmiz_1 :::"compound"::: ) ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v5_abcmiz_1 :::"compound"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" ))); end; theorem :: ABCMIZ_1:45 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool "ex" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set (Var "s1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "s2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 2))) & (Bool (Set (Var "a")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s1"))) & (Bool (Set (Var "b")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s2"))) ")" ))) "holds" (Bool (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k4_pre_poly :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_pre_poly :::"*>"::: ) )) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "o"))))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "o" be ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "C")); assume (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 2)) ; let "e1", "e2" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); assume (Bool "ex" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "C")) "st" (Bool "(" (Bool (Set (Var "s1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "s2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 2))) & (Bool (Set (Const "e1")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set (Var "s1"))) & (Bool (Set (Const "e2")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set (Var "s2"))) ")" )) ; func "o" :::"term"::: "(" "e1" "," "e2" ")" -> ($#m1_subset_1 :::"expression":::) "of" "C" equals :: ABCMIZ_1:def 31 (Set (Set ($#k4_tarski :::"["::: ) "o" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k4_pre_poly :::"<*"::: ) "e1" "," "e2" ($#k4_pre_poly :::"*>"::: ) )); end; :: deftheorem defines :::"term"::: ABCMIZ_1:def 31 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" )) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool "ex" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set (Var "s1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "s2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "o")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 2))) & (Bool (Set (Var "e1")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s1"))) & (Bool (Set (Var "e2")) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s2"))) ")" ))) "holds" (Bool (Set (Set (Var "o")) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "e1")) "," (Set (Var "e2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k4_pre_poly :::"<*"::: ) (Set (Var "e1")) "," (Set (Var "e2")) ($#k4_pre_poly :::"*>"::: ) )))))); theorem :: ABCMIZ_1:46 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")))) & (Bool (Set (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k4_pre_poly :::"<*"::: ) (Set (Var "a")) "," (Set (Var "t")) ($#k4_pre_poly :::"*>"::: ) ))) ")" )))) ; theorem :: ABCMIZ_1:47 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t1")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "b")) "," (Set (Var "t2")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Var "t2"))) ")" )))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); let "t" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); cluster (Set (Set "(" ($#k16_abcmiz_1 :::"ast"::: ) "C" ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" "a" "," "t" ")" ) -> ($#v5_abcmiz_1 :::"compound"::: ) ; end; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "s" be ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "S")); assume (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "S")) "st" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Const "s")))) ; mode :::"OperSymbol"::: "of" "s" -> ($#m1_subset_1 :::"OperSymbol":::) "of" "S" means :: ABCMIZ_1:def 32 (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) it) ($#r1_hidden :::"="::: ) "s"); end; :: deftheorem defines :::"OperSymbol"::: ABCMIZ_1:def 32 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) "st" (Bool (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S")) "st" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Var "s"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_abcmiz_1 :::"OperSymbol"::: ) "of" (Set (Var "s"))) "iff" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )))); definitionlet "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); :: original: :::"non_op"::: redefine func :::"non_op"::: "C" -> ($#m2_abcmiz_1 :::"OperSymbol"::: ) "of" (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); :: original: :::"ast"::: redefine func :::"ast"::: "C" -> ($#m2_abcmiz_1 :::"OperSymbol"::: ) "of" (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); end; theorem :: ABCMIZ_1:48 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "s1")) ($#r1_hidden :::"<>"::: ) (Set (Var "s2")))) "holds" (Bool "for" (Set (Var "t1")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s1")) (Bool "for" (Set (Var "t2")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set (Var "s2")) "holds" (Bool (Set (Var "t1")) ($#r1_hidden :::"<>"::: ) (Set (Var "t2"))))))) ; begin definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"QuasiTerms"::: "C" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" ) equals :: ABCMIZ_1:def 33 (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) "C" ")" )); end; :: deftheorem defines :::"QuasiTerms"::: ABCMIZ_1:def 33 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) (Set (Var "C")) ")" )))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_trees_3 :::"constituted-DTrees"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); mode quasi-term of "C" is ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k14_abcmiz_1 :::"a_Term"::: ) "C"); end; theorem :: ABCMIZ_1:49 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set (Var "z")) "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C"))) "iff" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C")))) ")" ))) ; definitionlet "x" be ($#m1_subset_1 :::"variable":::); let "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); func "x" :::"-term"::: "C" -> ($#m1_abcmiz_1 :::"quasi-term":::) "of" "C" equals :: ABCMIZ_1:def 34 (Set ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) "x" "," (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k4_tarski :::"]"::: ) )); end; :: deftheorem defines :::"-term"::: ABCMIZ_1:def 34 : (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set ($#k8_abcmiz_1 :::"a_Term"::: ) ) ($#k4_tarski :::"]"::: ) ))))); theorem :: ABCMIZ_1:50 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"variable":::) (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "st" (Bool (Bool (Set (Set (Var "x1")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x2")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C2"))))) "holds" (Bool (Set (Var "x1")) ($#r1_hidden :::"="::: ) (Set (Var "x2"))))) ; registrationlet "x" be ($#m1_subset_1 :::"variable":::); let "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set "x" ($#k35_abcmiz_1 :::"-term"::: ) "C") -> ($#~v5_abcmiz_1 "non" ($#v5_abcmiz_1 :::"compound"::: ) ) ; end; theorem :: ABCMIZ_1:51 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#v6_trees_3 :::"DTree-yielding"::: ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C")) ")" ) ($#k3_finseq_2 :::"*"::: ) )) ")" ) ")" )))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "c" be ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Const "C")); let "p" be ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Const "C"))); assume (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Const "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Const "c")) ")" ))) ; func "c" :::"-trm"::: "p" -> ($#v5_abcmiz_1 :::"compound"::: ) ($#m1_subset_1 :::"expression":::) "of" "C" equals :: ABCMIZ_1:def 35 (Set (Set ($#k4_tarski :::"["::: ) "c" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) "p"); end; :: deftheorem defines :::"-trm"::: ABCMIZ_1:def 35 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))))))); theorem :: ABCMIZ_1:52 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "c"))))))) ; theorem :: ABCMIZ_1:53 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C"))))) "or" (Bool "ex" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C"))(Bool "ex" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")))) ")" ))) "or" (Bool "ex" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))))) "or" (Bool "ex" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))(Bool "ex" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" )))) ")" ))) ; theorem :: ABCMIZ_1:54 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")))))))) ; theorem :: ABCMIZ_1:55 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ))))))) ; theorem :: ABCMIZ_1:56 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "b")) "," (Set (Var "t")) ")" ))))) ; theorem :: ABCMIZ_1:57 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Set (Var "e")) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_abcmiz_1 :::"non_op"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a"))))))) ; theorem :: ABCMIZ_1:58 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Set (Var "e")) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k9_abcmiz_1 :::"*"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))(Bool "ex" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" )))))) ; begin definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); func :::"Non"::: "a" -> ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") equals :: ABCMIZ_1:def 36 (Set "a" ($#k5_trees_2 :::"|"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k3_pre_poly :::"*>"::: ) )) if (Bool "ex" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") "st" (Bool "a" ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9"))))) otherwise (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) "a"); end; :: deftheorem defines :::"Non"::: ABCMIZ_1:def 36 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool "(" "(" (Bool (Bool "ex" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))))) "implies" (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_trees_2 :::"|"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k3_pre_poly :::"*>"::: ) ))) ")" & "(" (Bool (Bool "(" "for" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9"))))) ")" )) "implies" (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")))) ")" ")" ))); definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); attr "a" is :::"positive"::: means :: ABCMIZ_1:def 37 (Bool "for" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") "holds" (Bool (Bool "not" "a" ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))))); end; :: deftheorem defines :::"positive"::: ABCMIZ_1:def 37 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ) "iff" (Bool "for" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))))) ")" ))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v6_abcmiz_1 :::"positive"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; theorem :: ABCMIZ_1:59 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); attr "a" is :::"negative"::: means :: ABCMIZ_1:def 38 (Bool "ex" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") "st" (Bool "(" (Bool (Set (Var "a9")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ) & (Bool "a" ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))) ")" )); end; :: deftheorem defines :::"negative"::: ABCMIZ_1:def 38 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v7_abcmiz_1 :::"negative"::: ) ) "iff" (Bool "ex" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "a9")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))) ")" )) ")" ))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); cluster (Set ($#k37_abcmiz_1 :::"Non"::: ) "a") -> ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#v7_abcmiz_1 :::"negative"::: ) ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#v7_abcmiz_1 :::"negative"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; theorem :: ABCMIZ_1:60 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "ex" (Set (Var "a9")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))) & (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a9"))) ")" )))) ; theorem :: ABCMIZ_1:61 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#v7_abcmiz_1 :::"negative"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "ex" (Set (Var "a9")) "being" ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a9")))) & (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a9"))) ")" )))) ; theorem :: ABCMIZ_1:62 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set "(" ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#v7_abcmiz_1 :::"negative"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); cluster (Set ($#k37_abcmiz_1 :::"Non"::: ) "a") -> ($#v6_abcmiz_1 :::"positive"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); attr "a" is :::"regular"::: means :: ABCMIZ_1:def 39 (Bool "(" (Bool "a" "is" ($#v6_abcmiz_1 :::"positive"::: ) ) "or" (Bool "a" "is" ($#v7_abcmiz_1 :::"negative"::: ) ) ")" ); end; :: deftheorem defines :::"regular"::: ABCMIZ_1:def 39 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v8_abcmiz_1 :::"regular"::: ) ) "iff" (Bool "(" (Bool (Set (Var "a")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ) "or" (Bool (Set (Var "a")) "is" ($#v7_abcmiz_1 :::"negative"::: ) ) ")" ) ")" ))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#v6_abcmiz_1 :::"positive"::: ) -> ($#~v7_abcmiz_1 "non" ($#v7_abcmiz_1 :::"negative"::: ) ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); cluster ($#v7_abcmiz_1 :::"negative"::: ) -> ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"QuasiAdjs"::: "C" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "C" "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) ")" ")" ) equals :: ABCMIZ_1:def 40 "{" (Set (Var "a")) where a "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") : (Bool (Set (Var "a")) "is" ($#v8_abcmiz_1 :::"regular"::: ) ) "}" ; end; :: deftheorem defines :::"QuasiAdjs"::: ABCMIZ_1:def 40 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) : (Bool (Set (Var "a")) "is" ($#v8_abcmiz_1 :::"regular"::: ) ) "}" )); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_trees_3 :::"constituted-DTrees"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); mode quasi-adjective of "C" is ($#v8_abcmiz_1 :::"regular"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; theorem :: ABCMIZ_1:63 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set (Var "z")) "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C"))) "iff" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")))) ")" ))) ; theorem :: ABCMIZ_1:64 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set (Var "z")) "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C"))) "iff" (Bool "(" (Bool (Set (Var "z")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) "or" (Bool (Set (Var "z")) "is" ($#v7_abcmiz_1 :::"negative"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) ")" ) ")" ))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v6_abcmiz_1 "non" ($#v6_abcmiz_1 :::"positive"::: ) ) ($#v8_abcmiz_1 :::"regular"::: ) -> ($#v7_abcmiz_1 :::"negative"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); cluster ($#~v7_abcmiz_1 "non" ($#v7_abcmiz_1 :::"negative"::: ) ) ($#v8_abcmiz_1 :::"regular"::: ) -> ($#v6_abcmiz_1 :::"positive"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v6_abcmiz_1 :::"positive"::: ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v7_abcmiz_1 :::"negative"::: ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; theorem :: ABCMIZ_1:65 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) (Bool "ex" (Set (Var "v")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) & (Bool "ex" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "v")) ")" ))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")))) ")" )) ")" )))) ; theorem :: ABCMIZ_1:66 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "v")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "v")) ")" )))) "holds" (Bool (Set (Set (Var "v")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) "is" ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")))))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "a" be ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Const "C")); cluster (Set ($#k37_abcmiz_1 :::"Non"::: ) "a") -> ($#v8_abcmiz_1 :::"regular"::: ) ; end; theorem :: ABCMIZ_1:67 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set "(" ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ; theorem :: ABCMIZ_1:68 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a1"))) ($#r1_hidden :::"="::: ) (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a2"))))) "holds" (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Var "a2"))))) ; theorem :: ABCMIZ_1:69 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set (Var "a"))))) ; begin definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "q" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); attr "q" is :::"pure"::: means :: ABCMIZ_1:def 41 (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C") "holds" (Bool (Bool "not" "q" ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) "C" ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ))))); end; :: deftheorem defines :::"pure"::: ABCMIZ_1:def 41 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "q")) "is" ($#v9_abcmiz_1 :::"pure"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool (Bool "not" (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ))))) ")" ))); theorem :: ABCMIZ_1:70 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k6_abcmiz_1 :::"a_Type"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "m")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "t")) "is" ($#v9_abcmiz_1 :::"pure"::: ) ) ")" )))) ; theorem :: ABCMIZ_1:71 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k7_abcmiz_1 :::"an_Adj"::: ) )) & (Bool (Set ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a")) "is" ($#v6_abcmiz_1 :::"positive"::: ) ) ")" )))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v9_abcmiz_1 :::"pure"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"QuasiTypes"::: "C" -> ($#m1_hidden :::"set"::: ) equals :: ABCMIZ_1:def 42 "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "t")) ($#k1_domain_1 :::"]"::: ) ) where t "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"), A "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) "C" ")" ) : (Bool (Set (Var "t")) "is" ($#v9_abcmiz_1 :::"pure"::: ) ) "}" ; end; :: deftheorem defines :::"QuasiTypes"::: ABCMIZ_1:def 42 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "t")) ($#k1_domain_1 :::"]"::: ) ) where t "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))), A "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) : (Bool (Set (Var "t")) "is" ($#v9_abcmiz_1 :::"pure"::: ) ) "}" )); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); mode :::"quasi-type"::: "of" "C" -> ($#m1_hidden :::"set"::: ) means :: ABCMIZ_1:def 43 (Bool it ($#r2_hidden :::"in"::: ) (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) "C")); end; :: deftheorem defines :::"quasi-type"::: ABCMIZ_1:def 43 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C"))) "iff" (Bool (Set (Var "b2")) ($#r2_hidden :::"in"::: ) (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C")))) ")" ))); theorem :: ABCMIZ_1:72 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set (Var "z")) "is" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C"))) "iff" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" )(Bool "ex" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "q")) ($#k1_domain_1 :::"]"::: ) )))) ")" ))) ; theorem :: ABCMIZ_1:73 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) "is" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C"))) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" )) & (Bool (Set (Var "y")) "is" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")))) ")" ) ")" ))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster -> ($#v1_xtuple_0 :::"pair"::: ) for ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C"; end; theorem :: ABCMIZ_1:74 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "ex" (Set (Var "m")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool "(" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")))) & (Bool "ex" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "m")) ")" ))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")))) ")" )) ")" )))) ; theorem :: ABCMIZ_1:75 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "m")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "m")) ")" )))) "holds" (Bool (Set (Set (Var "m")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p"))) "is" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))))))) ; theorem :: ABCMIZ_1:76 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) "holds" (Bool "(" (Bool (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")))) & (Bool (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C")))) & (Bool (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")))) ")" )) ; theorem :: ABCMIZ_1:77 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "e")) "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")))) "implies" (Bool "not" (Bool (Set (Var "e")) "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")))) ")" & "(" (Bool (Bool (Set (Var "e")) "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")))) "implies" (Bool "not" (Bool (Set (Var "e")) "is" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")))) ")" & "(" (Bool (Bool (Set (Var "e")) "is" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")))) "implies" (Bool "not" (Bool (Set (Var "e")) "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")))) ")" ")" ))) ; notationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "A" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Const "C")) ")" ); let "q" be ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); synonym "A" :::"ast"::: "q" for :::"[":::"C" "," "A":::"]":::; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "A" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Const "C")) ")" ); let "q" be ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); :: original: :::"ast"::: redefine func "A" :::"ast"::: "q" -> ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C"; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); cluster (Set ($#k1_xtuple_0 :::"vars"::: ) "T") -> ($#v1_finset_1 :::"finite"::: ) ; end; notationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); synonym :::"adjs"::: "T" for "C" :::"`1"::: ; synonym :::"the_base_of"::: "T" for "C" :::"`2"::: ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); :: original: :::"vars"::: redefine func :::"adjs"::: "T" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) "C" ")" ); :: original: :::"the_base_of"::: redefine func :::"the_base_of"::: "T" -> ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); end; theorem :: ABCMIZ_1:78 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set ($#k41_abcmiz_1 :::"adjs"::: ) (Set "(" (Set (Var "A")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set "(" (Set (Var "A")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" )))) ; theorem :: ABCMIZ_1:79 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set (Set (Var "A1")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A2")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q2"))))) "holds" (Bool "(" (Bool (Set (Var "A1")) ($#r1_hidden :::"="::: ) (Set (Var "A2"))) & (Bool (Set (Var "q1")) ($#r1_hidden :::"="::: ) (Set (Var "q2"))) ")" )))) ; theorem :: ABCMIZ_1:80 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set (Var "T")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ) ($#k40_abcmiz_1 :::"ast"::: ) (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")) ")" ))))) ; theorem :: ABCMIZ_1:81 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T1"))) ($#r1_hidden :::"="::: ) (Set ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T2")))) & (Bool (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T1"))) ($#r1_hidden :::"="::: ) (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T2"))))) "holds" (Bool (Set (Var "T1")) ($#r1_hidden :::"="::: ) (Set (Var "T2"))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); let "a" be ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Const "C")); func "a" :::"ast"::: "T" -> ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C" equals :: ABCMIZ_1:def 44 (Set ($#k4_tarski :::"["::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) "a" ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) "T" ")" ) ")" ) "," (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) "T" ")" ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"ast"::: ABCMIZ_1:def 44 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ) ")" ) "," (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")) ")" ) ($#k4_tarski :::"]"::: ) ))))); theorem :: ABCMIZ_1:82 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set ($#k41_abcmiz_1 :::"adjs"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ))) & (Bool (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")))) ")" )))) ; theorem :: ABCMIZ_1:83 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T"))))))) ; theorem :: ABCMIZ_1:84 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set "(" (Set (Var "b")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k43_abcmiz_1 :::"ast"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )))))) ; begin registrationlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "s" be ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "S")); let "X" be bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "t" be ($#m1_dtconstr :::"Term":::) "of" (Set (Const "S")) "," (Set (Const "X")); cluster (Set (Set "(" ($#k4_msafree3 :::"variables_in"::: ) "t" ")" ) ($#k1_funct_1 :::"."::: ) "s") -> ($#v1_finset_1 :::"finite"::: ) ; end; registrationlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "s" be ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "S")); let "X" be bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "t" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Const "S")) "," (Set (Const "X")) ")" ")" ); cluster (Set (Set "(" "S" ($#k2_msafree3 :::"variables_in"::: ) "t" ")" ) ($#k1_funct_1 :::"."::: ) "s") -> ($#v1_finset_1 :::"finite"::: ) ; end; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "s" be ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Const "S")); func "(" "X" "," "s" ")" :::"variables_in"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" )) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "X" ($#k1_funct_1 :::"."::: ) "s" ")" ) ")" ) means :: ABCMIZ_1:def 45 (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "S" ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t")) ")" ) ($#k1_funct_1 :::"."::: ) "s"))); end; :: deftheorem defines :::"variables_in"::: ABCMIZ_1:def 45 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" bbbadV3_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "X")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "X")) "," (Set (Var "s")) ")" ($#k44_abcmiz_1 :::"variables_in"::: ) )) "iff" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))) ")" ))))); definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "e" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); func :::"variables_in"::: "e" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) equals :: ABCMIZ_1:def 46 (Set (Set "(" "C" ($#k2_msafree3 :::"variables_in"::: ) "e" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) "C" ")" )); end; :: deftheorem defines :::"variables_in"::: ABCMIZ_1:def 46 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "C")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "e")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) (Set (Var "C")) ")" ))))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "e" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); cluster (Set ($#k45_abcmiz_1 :::"variables_in"::: ) "e") -> ($#v1_finset_1 :::"finite"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "e" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); func :::"vars"::: "e" -> ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) equals :: ABCMIZ_1:def 47 (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) "e" ")" )); end; :: deftheorem defines :::"vars"::: ABCMIZ_1:def 47 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "e")) ")" ))))); theorem :: ABCMIZ_1:85 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "e")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "e")))))) ; theorem :: ABCMIZ_1:86 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: ABCMIZ_1:87 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_abcmiz_1 :::"vars"::: ) (Set (Var "x")) ")" ))))) ; theorem :: ABCMIZ_1:88 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#v6_trees_3 :::"DTree-yielding"::: ) ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "t")) ")" ) where t "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")) : (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) "}" )))))) ; theorem :: ABCMIZ_1:89 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#v6_trees_3 :::"DTree-yielding"::: ) ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "t")) ")" ) where t "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")) : (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) "}" )))))) ; theorem :: ABCMIZ_1:90 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "t")) ")" ) where t "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")) : (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p")))) "}" ))))) ; theorem :: ABCMIZ_1:91 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "t")) ")" ) where t "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Var "C")) : (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p")))) "}" ))))) ; theorem :: ABCMIZ_1:92 (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set "(" (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" )) ($#r6_pboole :::"="::: ) (Set ($#k1_pboole :::"[[0]]"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))))))) ; theorem :: ABCMIZ_1:93 (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_hidden :::"DecoratedTree":::) "holds" (Bool (Set (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set "(" (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "t")) ($#k9_finseq_1 :::"*>"::: ) ) ")" )) ($#r6_pboole :::"="::: ) (Set (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t"))))))) ; theorem :: ABCMIZ_1:94 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "a")))))) ; theorem :: ABCMIZ_1:95 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "a")))))) ; theorem :: ABCMIZ_1:96 (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_hidden :::"DecoratedTree":::) "holds" (Bool (Set (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set "(" (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "t1")) "," (Set (Var "t2")) ($#k10_finseq_1 :::"*>"::: ) ) ")" )) ($#r6_pboole :::"="::: ) (Set (Set "(" (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t1")) ")" ) ($#k2_pboole :::"\/"::: ) (Set "(" (Set (Var "S")) ($#k2_msafree3 :::"variables_in"::: ) (Set (Var "t2")) ")" )))))) ; theorem :: ABCMIZ_1:97 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "a")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_1:98 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "a")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_1:99 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "a")))))) ; theorem :: ABCMIZ_1:100 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set "(" ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "a")))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); func :::"variables_in"::: "T" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) equals :: ABCMIZ_1:def 48 (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set "(" (Set "(" "(" (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) "C" ")" ) "," (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) "C" ")" ) ")" ($#k44_abcmiz_1 :::"variables_in"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) "T" ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) "T" ")" ) ")" )); end; :: deftheorem defines :::"variables_in"::: ABCMIZ_1:def 48 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set "(" (Set "(" "(" (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) (Set (Var "C")) ")" ) ")" ($#k44_abcmiz_1 :::"variables_in"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")) ")" ) ")" ))))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); cluster (Set ($#k47_abcmiz_1 :::"variables_in"::: ) "T") -> ($#v1_finset_1 :::"finite"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); func :::"vars"::: "T" -> ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) equals :: ABCMIZ_1:def 49 (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k47_abcmiz_1 :::"variables_in"::: ) "T" ")" )); end; :: deftheorem defines :::"vars"::: ABCMIZ_1:def 49 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k47_abcmiz_1 :::"variables_in"::: ) (Set (Var "T")) ")" ))))); theorem :: ABCMIZ_1:101 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set "(" ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T")))))) ; theorem :: ABCMIZ_1:102 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "a")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k47_abcmiz_1 :::"variables_in"::: ) (Set (Var "T")) ")" )))))) ; theorem :: ABCMIZ_1:103 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "a")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T")) ")" )))))) ; theorem :: ABCMIZ_1:104 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "holds" (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set (Var "A")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "a")) ")" ) where a "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "}" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "q")) ")" )))))) ; theorem :: ABCMIZ_1:105 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "holds" (Bool (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set "(" (Set (Var "A")) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "a")) ")" ) where a "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "}" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k46_abcmiz_1 :::"vars"::: ) (Set (Var "q")) ")" )))))) ; theorem :: ABCMIZ_1:106 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "q")) "being" ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) (Set "(" (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) ")" ) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "q")))))) ; theorem :: ABCMIZ_1:107 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "e")) "is" ($#v4_abcmiz_1 :::"ground"::: ) ) "iff" (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); attr "T" is :::"ground"::: means :: ABCMIZ_1:def 50 (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) "T") ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"ground"::: ABCMIZ_1:def 50 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v10_abcmiz_1 :::"ground"::: ) ) "iff" (Bool (Set ($#k47_abcmiz_1 :::"variables_in"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v4_abcmiz_1 :::"ground"::: ) ($#v9_abcmiz_1 :::"pure"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); cluster ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v3_trees_2 :::"DecoratedTree-like"::: ) ($#v4_abcmiz_1 :::"ground"::: ) ($#v8_abcmiz_1 :::"regular"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; theorem :: ABCMIZ_1:108 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#v4_abcmiz_1 :::"ground"::: ) ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) ")" ) ($#k40_abcmiz_1 :::"ast"::: ) (Set (Var "t"))) "is" ($#v10_abcmiz_1 :::"ground"::: ) ))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "t" be ($#v4_abcmiz_1 :::"ground"::: ) ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); cluster (Set "t" ($#k4_tarski :::"ast"::: ) ) -> ($#v10_abcmiz_1 :::"ground"::: ) for ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C"; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#v1_xtuple_0 :::"pair"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v10_abcmiz_1 :::"ground"::: ) for ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C"; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#v10_abcmiz_1 :::"ground"::: ) ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); let "a" be ($#v4_abcmiz_1 :::"ground"::: ) ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Const "C")); cluster (Set "a" ($#k43_abcmiz_1 :::"ast"::: ) "T") -> ($#v10_abcmiz_1 :::"ground"::: ) ; end; begin definitionfunc :::"VarPoset"::: -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#l1_orders_2 :::"Poset":::) equals :: ABCMIZ_1:def 51 (Set (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) "{" (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) where A "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) : (Bool verum) "}" ")" ) ($#k7_lattice3 :::"opp"::: ) ); end; :: deftheorem defines :::"VarPoset"::: ABCMIZ_1:def 51 : (Bool (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) "{" (Set "(" ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "A")) ")" ) where A "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) : (Bool verum) "}" ")" ) ($#k7_lattice3 :::"opp"::: ) )); theorem :: ABCMIZ_1:109 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set (Var "x"))) ")" )) ; theorem :: ABCMIZ_1:110 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) )) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) )) & (Bool (Set ($#k1_abcmiz_1 :::"varcl"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" )) ; registration cluster (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ; end; theorem :: ABCMIZ_1:111 (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "V1")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "V2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "V1")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "V2")))) & (Bool (Set (Set (Var "V1")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "V2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "V1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "V2")))) ")" )) ; registrationlet "V1", "V2" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ); identify ; identify ; end; theorem :: ABCMIZ_1:112 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) )) & (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "X")))) ")" )) ; registration cluster (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ; end; theorem :: ABCMIZ_1:113 (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); func :::"vars-function"::: "C" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k39_abcmiz_1 :::"QuasiTypes"::: ) "C" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) )) means :: ABCMIZ_1:def 52 (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C" "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T"))))); end; :: deftheorem defines :::"vars-function"::: ABCMIZ_1:def 52 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) )) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k50_abcmiz_1 :::"vars-function"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k48_abcmiz_1 :::"vars"::: ) (Set (Var "T"))))) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); attr "L" is :::"smooth"::: means :: ABCMIZ_1:def 53 (Bool "ex" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::)(Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "L" "," (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#r1_tarski :::"c="::: ) (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k50_abcmiz_1 :::"vars-function"::: ) (Set (Var "C")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L"))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set (Var "f")) ($#r4_waybel_0 :::"preserves_sup_of"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k7_domain_1 :::"}"::: ) )) ")" ) ")" ))); end; :: deftheorem defines :::"smooth"::: ABCMIZ_1:def 53 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v11_abcmiz_1 :::"smooth"::: ) ) "iff" (Bool "ex" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::)(Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set ($#k49_abcmiz_1 :::"VarPoset"::: ) ) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set ($#k39_abcmiz_1 :::"QuasiTypes"::: ) (Set (Var "C")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k50_abcmiz_1 :::"vars-function"::: ) (Set (Var "C")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "f")) ($#r4_waybel_0 :::"preserves_sup_of"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k7_domain_1 :::"}"::: ) )) ")" ) ")" ))) ")" )); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "T" be ($#v10_abcmiz_1 :::"ground"::: ) ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); cluster (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set ($#k1_tarski :::"{"::: ) "T" ($#k1_tarski :::"}"::: ) ) "," (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_tarski :::"{"::: ) "T" ($#k1_tarski :::"}"::: ) ) ")" ) "#)" ) -> ($#v11_abcmiz_1 :::"smooth"::: ) ; end; begin scheme :: ABCMIZ_1:sch 2 StructInd{ F1() -> ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::), P1[ ($#m1_hidden :::"set"::: ) ], F2() -> ($#m1_subset_1 :::"expression":::) "of" (Set F1 "(" ")" ) } : (Bool P1[(Set F2 "(" ")" )]) provided (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool P1[(Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set F1 "(" ")" ))])) and (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool "(" "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool P1[(Set (Var "t"))]) ")" )) "holds" (Bool P1[(Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")))]))) and (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool P1[(Set (Var "a"))])) "holds" (Bool P1[(Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set F1 "(" ")" ) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")))])) and (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool P1[(Set (Var "a"))])) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool P1[(Set (Var "t"))])) "holds" (Bool P1[(Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set F1 "(" ")" ) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" )]))) proof end; definitionlet "S" be ($#l1_msualg_1 :::"ManySortedSign"::: ) ; attr "S" is :::"with_an_operation_for_each_sort"::: means :: ABCMIZ_1:def 54 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" "S"))); let "X" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); attr "X" is :::"with_missing_variables"::: means :: ABCMIZ_1:def 55 (Bool (Set "X" ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" "S"))); end; :: deftheorem defines :::"with_an_operation_for_each_sort"::: ABCMIZ_1:def 54 : (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v12_abcmiz_1 :::"with_an_operation_for_each_sort"::: ) ) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" (Set (Var "S"))))) ")" )); :: deftheorem defines :::"with_missing_variables"::: ABCMIZ_1:def 55 : (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ) "iff" (Bool (Set (Set (Var "X")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u2_msualg_1 :::"ResultSort"::: ) "of" (Set (Var "S"))))) ")" ))); theorem :: ABCMIZ_1:114 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set (Var "X")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S")) "st" (Bool (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Var "s"))))) ")" ))) ; registration cluster (Set ($#k27_abcmiz_1 :::"MaxConstrSign"::: ) ) -> ($#v1_msualg_1 :::"strict"::: ) ($#v12_abcmiz_1 :::"with_an_operation_for_each_sort"::: ) ; let "C" be ($#l1_msualg_1 :::"ConstructorSignature":::); cluster (Set ($#k28_abcmiz_1 :::"MSVars"::: ) "C") -> ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ; end; registrationlet "S" be ($#l1_msualg_1 :::"ManySortedSign"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) bbbadV2_RELAT_1() (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) -> ($#v13_abcmiz_1 :::"with_missing_variables"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_msualg_1 :::"ManySortedSign"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v13_abcmiz_1 :::"with_missing_variables"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) bbbadV14_STRUCT_0() ($#v1_msualg_1 :::"strict"::: ) bbbadV1_INSTALG1() ($#v1_abcmiz_1 :::"constructor"::: ) ($#v3_abcmiz_1 :::"initialized"::: ) ($#v12_abcmiz_1 :::"with_an_operation_for_each_sort"::: ) for ($#l1_msualg_1 :::"ManySortedSign"::: ) ; end; registrationlet "C" be ($#v12_abcmiz_1 :::"with_an_operation_for_each_sort"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C") ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) -> ($#v13_abcmiz_1 :::"with_missing_variables"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_lang1 :::"DTConstrStr"::: ) ; :: original: :::"Terminals"::: redefine func :::"Terminals"::: "G" -> ($#m1_subset_1 :::"Subset":::) "of" "G"; :: original: :::"NonTerminals"::: redefine func :::"NonTerminals"::: "G" -> ($#m1_subset_1 :::"Subset":::) "of" "G"; end; theorem :: ABCMIZ_1:115 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_lang1 :::"DTConstrStr"::: ) "st" (Bool (Bool (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set (Var "D1"))) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set (Var "D2"))))) "holds" (Bool "(" (Bool (Set ($#k52_abcmiz_1 :::"NonTerminals"::: ) (Set (Var "D1"))) ($#r1_tarski :::"c="::: ) (Set ($#k52_abcmiz_1 :::"NonTerminals"::: ) (Set (Var "D2")))) & (Bool (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "D1"))) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D2")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D1")))) & "(" (Bool (Bool (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D1"))) ($#r1_tarski :::"c="::: ) (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D2"))))) "implies" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "D1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "D2")))) ")" ")" )) ; theorem :: ABCMIZ_1:116 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_lang1 :::"DTConstrStr"::: ) "st" (Bool (Bool (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D1"))) ($#r1_tarski :::"c="::: ) (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set (Var "D2")))) & (Bool (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set (Var "D1"))) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set (Var "D2"))))) "holds" (Bool (Set ($#k4_dtconstr :::"TS"::: ) (Set (Var "D1"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_dtconstr :::"TS"::: ) (Set (Var "D2"))))) ; theorem :: ABCMIZ_1:117 (Bool "for" (Set (Var "S")) "being" ($#l1_msualg_1 :::"ManySortedSign"::: ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r2_pboole :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "X")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) )) "holds" (Bool (Set (Var "Y")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ))) ; theorem :: ABCMIZ_1:118 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) ($#r2_pboole :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k3_msafree :::"coprod"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k3_msafree :::"coprod"::: ) (Set (Var "Y")) ")" ))))) ; theorem :: ABCMIZ_1:119 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r2_pboole :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" ))))) ; theorem :: ABCMIZ_1:120 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) )) "holds" (Bool "(" (Bool (Set ($#k52_abcmiz_1 :::"NonTerminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_tarski :::"{"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k3_msafree :::"coprod"::: ) (Set (Var "X")) ")" ))) ")" ))) ; theorem :: ABCMIZ_1:121 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r2_pboole :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "X")) "is" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) )) "holds" (Bool "(" (Bool (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_lang1 :::"Rules"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set ($#k4_dtconstr :::"TS"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k4_dtconstr :::"TS"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" ))) ")" ))) ; theorem :: ABCMIZ_1:122 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k51_abcmiz_1 :::"Terminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set "(" ($#k14_abcmiz_1 :::"a_Term"::: ) (Set (Var "C")) ")" ) ($#k1_domain_1 :::"]"::: ) ))) ")" ))) ; theorem :: ABCMIZ_1:123 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k52_abcmiz_1 :::"NonTerminals"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ))) "iff" (Bool "(" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) )) "or" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) )) "or" (Bool "ex" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) "st" (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#k4_tarski :::"]"::: ) ))) ")" ) ")" ))) ; theorem :: ABCMIZ_1:124 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ))))) "holds" (Bool (Set (Var "t")) "is" ($#m1_dtconstr :::"Term":::) "of" (Set (Var "S")) "," (Set "(" (Set (Var "X")) ($#k2_pboole :::"\/"::: ) (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k8_funcop_1 :::"-->"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ))))) ; theorem :: ABCMIZ_1:125 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m1_dtconstr :::"Term":::) "of" (Set (Var "S")) "," (Set "(" (Set (Var "X")) ($#k2_pboole :::"\/"::: ) (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k8_funcop_1 :::"-->"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ))))) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k7_msaterm :::"the_sort_of"::: ) (Set (Var "t")) ")" )))))) ; theorem :: ABCMIZ_1:126 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_lang1 :::"DTConstrStr"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "s")) ($#r1_lang1 :::"==>"::: ) (Set (Var "p")))) "holds" (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))))))) ; theorem :: ABCMIZ_1:127 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Symbol":::) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "g2")) "being" ($#m1_subset_1 :::"Symbol":::) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" ) (Bool "for" (Set (Var "p1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" )) (Bool "for" (Set (Var "p2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "Y")) ")" )) "st" (Bool (Bool (Set (Var "g1")) ($#r1_hidden :::"="::: ) (Set (Var "g2"))) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Var "g1")) ($#r1_lang1 :::"==>"::: ) (Set (Var "p1")))) "holds" (Bool (Set (Var "g2")) ($#r1_lang1 :::"==>"::: ) (Set (Var "p2"))))))))) ; theorem :: ABCMIZ_1:128 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#v13_abcmiz_1 :::"with_missing_variables"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ))) ($#r1_hidden :::"="::: ) (Set ($#k4_dtconstr :::"TS"::: ) (Set "(" ($#k5_msafree :::"DTConMSA"::: ) (Set (Var "X")) ")" ))))) ; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); mode :::"term-transformation"::: "of" "S" "," "X" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" )) ")" ) means :: ABCMIZ_1:def 56 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" "S" "holds" (Bool (Set it ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))); end; :: deftheorem defines :::"term-transformation"::: ABCMIZ_1:def 56 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Var "S")) "," (Set (Var "X"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "b3")) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))) ")" )))); theorem :: ABCMIZ_1:129 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Var "S")) "," (Set (Var "X"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")))))) ")" )))) ; theorem :: ABCMIZ_1:130 (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "f")) "being" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Var "S")) "," (Set (Var "X")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SortSymbol":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "p"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "s")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) ")" )))))) ; definitionlet "S" be ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::); let "X" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "t" be ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Const "S")) "," (Set (Const "X")); attr "t" is :::"substitution"::: means :: ABCMIZ_1:def 57 (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" "S" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" ) "st" (Bool (Bool (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" "S" "," "X" ")" ")" )))) & (Bool (Set (Var "q")) ($#r2_relset_1 :::"="::: ) (Set "t" ($#k1_partfun1 :::"*"::: ) (Set (Var "p"))))) "holds" (Bool (Set "t" ($#k1_funct_1 :::"."::: ) (Set "(" (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "q")))))); end; :: deftheorem defines :::"substitution"::: ABCMIZ_1:def 57 : (Bool "for" (Set (Var "S")) "being" ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#l1_msualg_1 :::"Signature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "t")) "being" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Var "S")) "," (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "t")) "is" ($#v14_abcmiz_1 :::"substitution"::: ) ) "iff" (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )))) & (Bool (Set (Var "q")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "t")) ($#k1_partfun1 :::"*"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "t")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k4_tarski :::"]"::: ) ) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "q")))))) ")" )))); scheme :: ABCMIZ_1:sch 3 StructDef{ F1() -> ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::), F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"expression":::) "of" (Set F1 "(" ")" ), F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"expression":::) "of" (Set F1 "(" ")" ), F4( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"expression":::) "of" (Set F1 "(" ")" ), F5( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"expression":::) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "f")) "being" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k28_abcmiz_1 :::"MSVars"::: ) (Set F1 "(" ")" )) "st" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set F1 "(" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "x")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k11_lang1 :::"*"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "c")) "," (Set (Var "q")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set F1 "(" ")" ) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ")" )) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set F1 "(" ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set F1 "(" ")" ) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set F5 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ))) ")" ) ")" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool (Set F2 "(" (Set (Var "x")) ")" ) "is" ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set F1 "(" ")" ))) and (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set F4 "(" (Set (Var "c")) "," (Set (Var "p")) ")" ) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k2_msualg_1 :::"the_result_sort_of"::: ) (Set (Var "c")))))) and (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) "holds" (Bool (Set F3 "(" (Set (Var "a")) ")" ) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )))) and (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set F1 "(" ")" )) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set F1 "(" ")" )) "holds" (Bool (Set F5 "(" (Set (Var "a")) "," (Set (Var "t")) ")" ) "is" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set F1 "(" ")" ))))) proof end; begin definitionlet "A" be ($#m1_hidden :::"set"::: ) ; let "x", "y" be ($#m1_hidden :::"set"::: ) ; let "a", "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "A")); :: original: :::"IFIN"::: redefine func :::"IFIN"::: "(" "x" "," "y" "," "a" "," "b" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" "A"; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); mode valuation of "C" is ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "," (Set "(" ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C" ")" ); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); attr "f" is :::"irrelevant"::: means :: ABCMIZ_1:def 58 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k35_abcmiz_1 :::"-term"::: ) "C")))); end; :: deftheorem defines :::"irrelevant"::: ABCMIZ_1:def 58 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v15_abcmiz_1 :::"irrelevant"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))))) ")" ))); notationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); antonym :::"relevant"::: "f" for :::"irrelevant"::: ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v15_abcmiz_1 :::"irrelevant"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "," (Set "(" ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "," (Set "(" ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ); func "C" :::"idval"::: "X" -> ($#m1_subset_1 :::"valuation":::) "of" "C" equals :: ABCMIZ_1:def 59 "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) "C" ")" ) ($#k1_domain_1 :::"]"::: ) ) where x "is" ($#m1_subset_1 :::"variable":::) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X") "}" ; end; :: deftheorem defines :::"idval"::: ABCMIZ_1:def 59 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k54_abcmiz_1 :::"idval"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" ) ($#k1_domain_1 :::"]"::: ) ) where x "is" ($#m1_subset_1 :::"variable":::) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "}" ))); theorem :: ABCMIZ_1:131 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "C")) ($#k54_abcmiz_1 :::"idval"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" (Set (Var "C")) ($#k54_abcmiz_1 :::"idval"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))) ")" ) ")" ))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ); cluster (Set "C" ($#k54_abcmiz_1 :::"idval"::: ) "X") -> ($#v2_funct_1 :::"one-to-one"::: ) ($#v15_abcmiz_1 :::"irrelevant"::: ) ; end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "X" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ); cluster (Set "C" ($#k54_abcmiz_1 :::"idval"::: ) "X") -> ($#v1_xboole_0 :::"empty"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); func "f" :::"#"::: -> ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" "C" "," (Set ($#k28_abcmiz_1 :::"MSVars"::: ) "C") means :: ABCMIZ_1:def 60 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "implies" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) "C" ")" )) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")))) "implies" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) "C" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) "C")) ")" ")" ) ")" ) & (Bool "(" "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" "C" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C") "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set it ($#k11_lang1 :::"*"::: ) (Set (Var "p"))))) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "q"))))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) "C" ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set "(" it ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C") (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) "C" ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) "C" ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set "(" it ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) "," (Set "(" it ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"#"::: ABCMIZ_1:def 60 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "b3")) "being" ($#m4_abcmiz_1 :::"term-transformation"::: ) "of" (Set (Var "C")) "," (Set ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k55_abcmiz_1 :::"#"::: ) )) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "implies" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) "implies" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))) ")" ")" ) ")" ) & (Bool "(" "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b3")) ($#k11_lang1 :::"*"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "q"))))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) "," (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ))) ")" ) ")" ) ")" )))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); cluster (Set "f" ($#k55_abcmiz_1 :::"#"::: ) ) -> ($#v14_abcmiz_1 :::"substitution"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "e" be ($#m1_subset_1 :::"expression":::) "of" (Set (Const "C")); func "e" :::"at"::: "f" -> ($#m1_subset_1 :::"expression":::) "of" "C" equals :: ABCMIZ_1:def 61 (Set (Set "(" "f" ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "e"); end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 61 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "e"))))))); definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "p" be ($#m1_hidden :::"FinSequence":::); assume (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Const "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Const "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Const "C")) ")" ) ")" ")" )))) ; func "p" :::"at"::: "f" -> ($#m1_hidden :::"FinSequence":::) equals :: ABCMIZ_1:def 62 (Set (Set "(" "f" ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) "p"); end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 62 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" ))))) "holds" (Bool (Set (Set (Var "p")) ($#k57_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "p"))))))); definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "p" be ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Const "C"))); :: original: :::"at"::: redefine func "p" :::"at"::: "f" -> ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) "C") equals :: ABCMIZ_1:def 63 (Set (Set "(" "f" ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k11_lang1 :::"*"::: ) "p"); end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 63 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) "holds" (Bool (Set (Set (Var "p")) ($#k58_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k55_abcmiz_1 :::"#"::: ) ")" ) ($#k11_lang1 :::"*"::: ) (Set (Var "p"))))))); theorem :: ABCMIZ_1:132 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C"))))))) ; theorem :: ABCMIZ_1:133 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))))) ; theorem :: ABCMIZ_1:134 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "c")) "being" ($#v2_abcmiz_1 :::"constructor"::: ) ($#m1_subset_1 :::"OperSymbol":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k34_abcmiz_1 :::"QuasiTerms"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_msualg_1 :::"the_arity_of"::: ) (Set (Var "c")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set (Var "p")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k36_abcmiz_1 :::"-trm"::: ) (Set "(" (Set (Var "p")) ($#k58_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ))))))) ; theorem :: ABCMIZ_1:135 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set (Var "a")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_abcmiz_1 :::"non_op"::: ) (Set (Var "C")) ")" ) ($#k30_abcmiz_1 :::"term"::: ) (Set "(" (Set (Var "a")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )))))) ; theorem :: ABCMIZ_1:136 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "t")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set (Var "a")) "," (Set (Var "t")) ")" ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k33_abcmiz_1 :::"ast"::: ) (Set (Var "C")) ")" ) ($#k31_abcmiz_1 :::"term"::: ) "(" (Set "(" (Set (Var "a")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "t")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ")" )))))) ; theorem :: ABCMIZ_1:137 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool (Set (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set "(" (Set (Var "C")) ($#k54_abcmiz_1 :::"idval"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "e")))))) ; theorem :: ABCMIZ_1:138 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_abcmiz_1 :::"Vars"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "C")) ($#k54_abcmiz_1 :::"idval"::: ) (Set (Var "X")) ")" ) ($#k55_abcmiz_1 :::"#"::: ) ) ($#r2_relset_1 :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" )) ")" ))))) ; theorem :: ABCMIZ_1:139 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "e")))))) ; theorem :: ABCMIZ_1:140 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "f")) ($#k55_abcmiz_1 :::"#"::: ) ) ($#r2_relset_1 :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "the" ($#u3_msualg_1 :::"Sorts"::: ) "of" (Set "(" ($#k1_msafree3 :::"Free"::: ) "(" (Set (Var "C")) "," (Set "(" ($#k28_abcmiz_1 :::"MSVars"::: ) (Set (Var "C")) ")" ) ")" ")" )) ")" ))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "t" be ($#m1_abcmiz_1 :::"quasi-term":::) "of" (Set (Const "C")); :: original: :::"at"::: redefine func "t" :::"at"::: "f" -> ($#m1_abcmiz_1 :::"quasi-term":::) "of" "C"; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "a" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); :: original: :::"at"::: redefine func "a" :::"at"::: "f" -> ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "a" be ($#v6_abcmiz_1 :::"positive"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); cluster (Set "a" ($#k56_abcmiz_1 :::"at"::: ) "f") -> ($#v6_abcmiz_1 :::"positive"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "a" be ($#v7_abcmiz_1 :::"negative"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Const "C"))); cluster (Set "a" ($#k56_abcmiz_1 :::"at"::: ) "f") -> ($#v7_abcmiz_1 :::"negative"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) "C"); end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "a" be ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Const "C")); :: original: :::"at"::: redefine func "a" :::"at"::: "f" -> ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" "C"; end; theorem :: ABCMIZ_1:141 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Var "C")) "," (Set ($#k13_abcmiz_1 :::"an_Adj"::: ) (Set (Var "C"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" ($#k37_abcmiz_1 :::"Non"::: ) (Set (Var "a")) ")" ) ($#k60_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k37_abcmiz_1 :::"Non"::: ) (Set "(" (Set (Var "a")) ($#k60_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "t" be ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); :: original: :::"at"::: redefine func "t" :::"at"::: "f" -> ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); end; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "t" be ($#v9_abcmiz_1 :::"pure"::: ) ($#m1_abcmiz_1 :::"expression"::: ) "of" (Set (Const "C")) "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) (Set (Const "C"))); cluster (Set "t" ($#k56_abcmiz_1 :::"at"::: ) "f") -> ($#v9_abcmiz_1 :::"pure"::: ) for ($#m1_abcmiz_1 :::"expression"::: ) "of" "C" "," (Set ($#k12_abcmiz_1 :::"a_Type"::: ) "C"); end; theorem :: ABCMIZ_1:142 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#v15_abcmiz_1 :::"irrelevant"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "ex" (Set (Var "g")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#v15_abcmiz_1 :::"irrelevant"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "st" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))) ")" ) ")" ))))) ; theorem :: ABCMIZ_1:143 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#v15_abcmiz_1 :::"irrelevant"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C"))))) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")))) ")" ) ")" )) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k45_abcmiz_1 :::"variables_in"::: ) (Set (Var "e"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "e")))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Const "C")) ")" ); func "A" :::"at"::: "f" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) "C" ")" ) equals :: ABCMIZ_1:def 64 "{" (Set "(" (Set (Var "a")) ($#k61_abcmiz_1 :::"at"::: ) "f" ")" ) where a "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" "C" : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "A") "}" ; end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 64 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "holds" (Bool (Set (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "a")) ($#k61_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) where a "is" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "}" )))); theorem :: ABCMIZ_1:144 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "a")) ($#k61_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k6_domain_1 :::"}"::: ) )))))) ; theorem :: ABCMIZ_1:145 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "B")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )))))) ; theorem :: ABCMIZ_1:146 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f"))))))) ; registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "A" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Const "C")) ")" ); cluster (Set "A" ($#k63_abcmiz_1 :::"at"::: ) "f") -> ($#v1_finset_1 :::"finite"::: ) ; end; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); let "T" be ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Const "C")); func "T" :::"at"::: "f" -> ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" "C" equals :: ABCMIZ_1:def 65 (Set (Set "(" (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) "T" ")" ) ($#k63_abcmiz_1 :::"at"::: ) "f" ")" ) ($#k40_abcmiz_1 :::"ast"::: ) (Set "(" (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) "T" ")" ) ($#k62_abcmiz_1 :::"at"::: ) "f" ")" )); end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 65 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool (Set (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k40_abcmiz_1 :::"ast"::: ) (Set "(" (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")) ")" ) ($#k62_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )))))); theorem :: ABCMIZ_1:147 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set ($#k41_abcmiz_1 :::"adjs"::: ) (Set "(" (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k41_abcmiz_1 :::"adjs"::: ) (Set (Var "T")) ")" ) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f")))) & (Bool (Set ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set "(" (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_abcmiz_1 :::"the_base_of"::: ) (Set (Var "T")) ")" ) ($#k62_abcmiz_1 :::"at"::: ) (Set (Var "f")))) ")" )))) ; theorem :: ABCMIZ_1:148 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "a")) "being" ($#m1_abcmiz_1 :::"quasi-adjective":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k43_abcmiz_1 :::"ast"::: ) (Set (Var "T")) ")" ) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k61_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k43_abcmiz_1 :::"ast"::: ) (Set "(" (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ))))))) ; definitionlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f", "g" be ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); func "f" :::"at"::: "g" -> ($#m1_subset_1 :::"valuation":::) "of" "C" means :: ABCMIZ_1:def 66 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "g" ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) "C" ")" ) ($#k59_abcmiz_1 :::"at"::: ) "f" ")" ) ($#k59_abcmiz_1 :::"at"::: ) "g")) ")" ) ")" ); end; :: deftheorem defines :::"at"::: ABCMIZ_1:def 66 : (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k65_abcmiz_1 :::"at"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"variable":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k35_abcmiz_1 :::"-term"::: ) (Set (Var "C")) ")" ) ($#k59_abcmiz_1 :::"at"::: ) (Set (Var "f")) ")" ) ($#k59_abcmiz_1 :::"at"::: ) (Set (Var "g")))) ")" ) ")" ) ")" ))); registrationlet "C" be ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::); let "f", "g" be ($#v15_abcmiz_1 :::"irrelevant"::: ) ($#m1_subset_1 :::"valuation":::) "of" (Set (Const "C")); cluster (Set "f" ($#k65_abcmiz_1 :::"at"::: ) "g") -> ($#v15_abcmiz_1 :::"irrelevant"::: ) ; end; theorem :: ABCMIZ_1:149 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"expression":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f1")) ")" ) ($#k56_abcmiz_1 :::"at"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "e")) ($#k56_abcmiz_1 :::"at"::: ) (Set "(" (Set (Var "f1")) ($#k65_abcmiz_1 :::"at"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: ABCMIZ_1:150 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k38_abcmiz_1 :::"QuasiAdjs"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f1")) ")" ) ($#k63_abcmiz_1 :::"at"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k63_abcmiz_1 :::"at"::: ) (Set "(" (Set (Var "f1")) ($#k65_abcmiz_1 :::"at"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: ABCMIZ_1:151 (Bool "for" (Set (Var "C")) "being" ($#v3_abcmiz_1 :::"initialized"::: ) ($#l1_msualg_1 :::"ConstructorSignature":::) (Bool "for" (Set (Var "T")) "being" ($#m3_abcmiz_1 :::"quasi-type"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"valuation":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "(" (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f1")) ")" ) ($#k64_abcmiz_1 :::"at"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "T")) ($#k64_abcmiz_1 :::"at"::: ) (Set "(" (Set (Var "f1")) ($#k65_abcmiz_1 :::"at"::: ) (Set (Var "f2")) ")" )))))) ;