:: AOFA_000 semantic presentation begin notationlet "x", "y" be ($#m1_hidden :::"set"::: ) ; antonym "x" :::"nin"::: "y" for "x" :::"in"::: "y"; end; theorem :: AOFA_000:1 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h")))))) ; registrationlet "x", "y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#v2_relat_1 :::"non-empty"::: ) ; end; registrationlet "p", "q" be ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set "p" ($#k7_finseq_1 :::"^"::: ) "q") -> ($#v2_relat_1 :::"non-empty"::: ) ; end; definitionlet "f" be ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; pred "x" :::"is_a_unity_wrt"::: "f" means :: AOFA_000:def 1 (Bool "for" (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) "or" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) ")" )) "holds" (Bool "(" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," "x" ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," "x" ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" )); end; :: deftheorem defines :::"is_a_unity_wrt"::: AOFA_000:def 1 : (Bool "for" (Set (Var "f")) "being" ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) (Set (Var "f"))) "iff" (Bool "for" (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) "or" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) ")" )) "holds" (Bool "(" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" )) ")" ))); definitionlet "f" be ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_hidden :::"Function":::); attr "f" is :::"associative"::: means :: AOFA_000:def 2 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) )))); attr "f" is :::"unital"::: means :: AOFA_000:def 3 (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "x")) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) "f")); end; :: deftheorem defines :::"associative"::: AOFA_000:def 2 : (Bool "for" (Set (Var "f")) "being" ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_aofa_000 :::"associative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) )))) ")" )); :: deftheorem defines :::"unital"::: AOFA_000:def 3 : (Bool "for" (Set (Var "f")) "being" ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_aofa_000 :::"unital"::: ) ) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "x")) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) (Set (Var "f")))) ")" )); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Z" be ($#m1_finseq_2 :::"FinSequenceSet"::: ) "of" (Set (Const "X")); let "y" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "Y")); :: original: :::"-->"::: redefine func "Z" :::"-->"::: "y" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "Y"; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set (Set "(" "n" ($#k4_finseq_2 :::"-tuples_on"::: ) "X" ")" ) ($#k2_funcop_1 :::"-->"::: ) "x") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "X"; end; theorem :: AOFA_000:2 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k19_margrel1 :::"arity"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "X")) ")" ) ($#k1_aofa_000 :::"-->"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n")))))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set (Set "(" "n" ($#k4_finseq_2 :::"-tuples_on"::: ) "X" ")" ) ($#k2_funcop_1 :::"-->"::: ) "x") -> ($#v2_margrel1 :::"homogeneous"::: ) "n" ($#v8_comput_1 :::"-ary"::: ) for ($#v2_margrel1 :::"homogeneous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "X"; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "X" ($#k3_finseq_2 :::"*"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#v1_aofa_000 :::"associative"::: ) ($#v2_aofa_000 :::"unital"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "X" ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "X" ($#k3_finseq_2 :::"*"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v8_comput_1 :::"-ary"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "X" ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "X" ($#k3_finseq_2 :::"*"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 3) ($#v8_comput_1 :::"-ary"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "X" ($#k3_finseq_2 :::"*"::: ) ")" ) "," "X" ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: AOFA_000:3 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_trees_3 :::"FinTrees"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "x")) "," (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "t")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "x")) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))))))) ; definitionlet "f", "g" be ($#m1_hidden :::"Function":::); let "X" be ($#m1_hidden :::"set"::: ) ; func "(" "f" "," "X" ")" :::"+*"::: "g" -> ($#m1_hidden :::"Function":::) equals :: AOFA_000:def 4 (Set "g" ($#k1_funct_4 :::"+*"::: ) (Set "(" "f" ($#k5_relat_1 :::"|"::: ) "X" ")" )); end; :: deftheorem defines :::"+*"::: AOFA_000:def 4 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "(" (Set (Var "f")) "," (Set (Var "X")) ")" ($#k2_aofa_000 :::"+*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ))))); theorem :: AOFA_000:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" "(" (Set (Var "f")) "," (Set (Var "X")) ")" ($#k2_aofa_000 :::"+*"::: ) (Set (Var "g")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: AOFA_000:5 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"nin"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set "(" "(" (Set (Var "f")) "," (Set (Var "X")) ")" ($#k2_aofa_000 :::"+*"::: ) (Set (Var "g")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; definitionlet "X", "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f", "g" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ); let "A" be ($#m1_hidden :::"set"::: ) ; :: original: :::"+*"::: redefine func "(" "f" "," "A" ")" :::"+*"::: "g" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," "Y" ")" ); end; definitionlet "X", "Y", "Z" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ); let "g" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "Y")) "," (Set (Const "Z")) ")" ); :: original: :::"*"::: redefine func "g" :::"*"::: "f" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," "Z" ")" ); end; definitionlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; func "f" :::"orbit"::: "x" -> ($#m1_hidden :::"set"::: ) equals :: AOFA_000:def 5 "{" (Set "(" (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) "x" ")" ) where n "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ))) "}" ; end; :: deftheorem defines :::"orbit"::: AOFA_000:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) where n "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ))) "}" ))); theorem :: AOFA_000:6 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x")))))) ; theorem :: AOFA_000:7 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x")))))) ; theorem :: AOFA_000:8 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x")))))) ; theorem :: AOFA_000:9 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "x")))))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); assume (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Const "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Const "f")))) ; let "A", "x" be ($#m1_hidden :::"set"::: ) ; func "(" "A" "," "x" ")" :::"iter"::: "f" -> ($#m1_hidden :::"Function":::) means :: AOFA_000:def 6 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool "(" "(" (Bool (Bool (Set "f" ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) "A")) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) "x") ")" & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"nin"::: ) "A") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) "A") ")" )) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" ) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"iter"::: AOFA_000:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "A")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "A")) "," (Set (Var "x")) ")" ($#k6_aofa_000 :::"iter"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"nin"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" ) ")" ) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_hidden :::"Function":::); assume (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Const "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Const "f")))) ; let "A" be ($#m1_hidden :::"set"::: ) ; let "g" be ($#m1_hidden :::"Function":::); func "(" "A" "," "g" ")" :::"iter"::: "f" -> ($#m1_hidden :::"Function":::) means :: AOFA_000:def 7 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool "(" "(" (Bool (Bool (Set "f" ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) "A")) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"nin"::: ) "A") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) "A") ")" )) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" "f" "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" ) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"iter"::: AOFA_000:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "b4")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "A")) "," (Set (Var "g")) ")" ($#k7_aofa_000 :::"iter"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"nin"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" ) ")" ) ")" ) ")" ) ")" )))); theorem :: AOFA_000:10 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set "(" "(" (Set (Var "A")) "," (Set (Var "g")) ")" ($#k7_aofa_000 :::"iter"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) & (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"nin"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" )))) ; theorem :: AOFA_000:11 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" "(" (Set (Var "A")) "," (Set (Var "g")) ")" ($#k7_aofa_000 :::"iter"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "(" (Set (Var "A")) "," (Set (Var "g")) ")" ($#k7_aofa_000 :::"iter"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ))))) ; theorem :: AOFA_000:12 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"nin"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" "(" (Set (Var "A")) "," (Set (Var "g")) ")" ($#k7_aofa_000 :::"iter"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "X")) ")" ); let "A" be ($#m1_hidden :::"set"::: ) ; let "g" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "X")) ")" ); :: original: :::"iter"::: redefine func "(" "A" "," "g" ")" :::"iter"::: "f" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," "X" ")" ); end; begin theorem :: AOFA_000:13 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k1_unialg_1 :::"signature"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "S"))) ")" )))) ; theorem :: AOFA_000:14 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k5_numbers :::"NAT"::: ) )) & (Bool (Set ($#k1_unialg_1 :::"signature"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "S"))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "S")))) & (Bool (Set (Var "j")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "j")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "i")))) ")" ) ")" ))) ; theorem :: AOFA_000:15 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "S")))) & (Bool (Set (Var "j")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "j")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "X")) ")" ) "," (Set (Var "X")) (Bool "ex" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k1_unialg_1 :::"signature"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "S"))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" )))))) ; registrationlet "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" "f" "," "D" ")" ")" )); end; registrationlet "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v3_trees_2 :::"DecoratedTree-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" "f" "," "D" ")" ")" )); cluster -> ($#v6_trees_3 :::"DTree-yielding"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" "f" "," "D" ")" ")" )); end; theorem :: AOFA_000:16 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_lang1 :::"DTConstrStr"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k4_dtconstr :::"TS"::: ) (Set (Var "G")))) "or" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Symbol":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k1_lang1 :::"Terminals"::: ) (Set (Var "G")))) & (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set ($#k2_trees_4 :::"root-tree"::: ) (Set (Var "d")))) ")" )) "or" (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"Symbol":::) "of" (Set (Var "G"))(Bool "ex" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k4_dtconstr :::"TS"::: ) (Set (Var "G"))) "st" (Bool "(" (Bool (Set (Var "o")) ($#r1_lang1 :::"==>"::: ) (Set ($#k1_dtconstr :::"roots"::: ) (Set (Var "p")))) & (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set (Set (Var "o")) ($#k8_trees_4 :::"-tree"::: ) (Set (Var "p")))) ")" ))) ")" ))) ; theorem :: AOFA_000:17 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "S"))))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" )) ")" ) ")" ")" ) "," (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set (Var "S")) "," (Set (Var "X")) ")" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p")))))))) ; definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; let "B" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); let "n" be ($#m1_hidden :::"Nat":::); func "B" :::"|^"::: "n" -> ($#m1_subset_1 :::"Subset":::) "of" "A" means :: AOFA_000:def 8 (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ")" ) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) "n")) & (Bool (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "B") & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," "A" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")) ")" ) where o "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A")), p "is" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k3_finseq_2 :::"*"::: ) ) : (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," "A" ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set (Var "n")))) ")" ) "}" )) ")" ) ")" )); end; :: deftheorem defines :::"|^"::: AOFA_000:def 8 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ")" ) "st" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")) ")" ) where o "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A")))), p "is" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k3_finseq_2 :::"*"::: ) ) : (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "F")) ($#k8_nat_1 :::"."::: ) (Set (Var "n")))) ")" ) "}" )) ")" ) ")" )) ")" ))))); theorem :: AOFA_000:18 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) ; theorem :: AOFA_000:19 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")) ")" ) where o "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A")))), p "is" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k3_finseq_2 :::"*"::: ) ) : (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" ) "}" ))))) ; theorem :: AOFA_000:20 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) "or" (Bool "ex" (Set (Var "o")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))))(Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k3_finseq_2 :::"*"::: ) ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" ))) ")" ) ")" ))))) ; theorem :: AOFA_000:21 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "m"))))))) ; theorem :: AOFA_000:22 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B1")) "," (Set (Var "B2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "B1")) ($#r1_tarski :::"c="::: ) (Set (Var "B2")))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "B1")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B2")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))))) ; theorem :: AOFA_000:23 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) "or" (Bool "ex" (Set (Var "o")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))))(Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k3_finseq_2 :::"*"::: ) ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" ))) ")" ) ")" ))))) ; scheme :: AOFA_000:sch 1 MaxVal{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "n"))]))) provided (Bool (Set F2 "(" ")" ) "is" ($#v1_finset_1 :::"finite"::: ) ) and (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool P1[(Set (Var "x")) "," (Set (Var "n"))]))) and (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool P1[(Set (Var "x")) "," (Set (Var "n"))]) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "m"))]))) proof end; theorem :: AOFA_000:24 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")) ")" ) where n "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" )) & (Bool (Set (Var "C")) "is" ($#v1_unialg_2 :::"opers_closed"::: ) ) ")" )))) ; theorem :: AOFA_000:25 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v1_unialg_2 :::"opers_closed"::: ) ) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")) ")" ) where n "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ) ($#r1_tarski :::"c="::: ) (Set (Var "C"))))) ; definitionlet "A" be ($#l1_unialg_1 :::"Universal_Algebra":::); func :::"Generators"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" "A" equals :: AOFA_000:def 9 (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "o")) ")" ) where o "is" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) "A") : (Bool verum) "}" ")" )); end; :: deftheorem defines :::"Generators"::: AOFA_000:def 9 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "holds" (Bool (Set ($#k10_aofa_000 :::"Generators"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "o")) ")" ) where o "is" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) (Set (Var "A"))) : (Bool verum) "}" ")" )))); theorem :: AOFA_000:26 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k10_aofa_000 :::"Generators"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "o")) "being" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) (Set (Var "A"))) "holds" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "o")))))) ")" ))) ; theorem :: AOFA_000:27 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v1_unialg_2 :::"opers_closed"::: ) )) "holds" (Bool (Set ($#k6_unialg_2 :::"Constants"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) ; theorem :: AOFA_000:28 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "st" (Bool (Bool (Set ($#k6_unialg_2 :::"Constants"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "A"))) "is" ($#v1_unialg_2 :::"opers_closed"::: ) )) ; theorem :: AOFA_000:29 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) "st" (Bool (Bool (Set ($#k6_unialg_2 :::"Constants"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Var "G")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: AOFA_000:30 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "G")) "is" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A"))) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))))) ")" ))) ; theorem :: AOFA_000:31 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "B")) "is" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")))))) ; theorem :: AOFA_000:32 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool "(" "for" (Set (Var "o")) "being" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) (Set (Var "A"))) "holds" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "o"))))) ")" )) "holds" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "G")))))) ; theorem :: AOFA_000:33 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) "holds" (Bool (Set ($#k10_aofa_000 :::"Generators"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Var "G"))))) ; theorem :: AOFA_000:34 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#v2_freealg :::"free"::: ) ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set ($#k10_aofa_000 :::"Generators"::: ) (Set (Var "A")))))) ; registrationlet "A" be ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::); cluster (Set ($#k10_aofa_000 :::"Generators"::: ) "A") -> ($#v2_freealg :::"free"::: ) for ($#m1_freealg :::"GeneratorSet"::: ) "of" "A"; end; definitionlet "A" be ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::); :: original: :::"Generators"::: redefine func :::"Generators"::: "A" -> ($#m1_freealg :::"GeneratorSet"::: ) "of" "A"; end; registrationlet "A", "B" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "B" ($#k2_zfmisc_1 :::":]"::: ) ) -> ($#v1_freealg :::"disjoint_with_NAT"::: ) ; end; theorem :: AOFA_000:35 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "B")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "st" (Bool (Bool (Set (Var "h1")) ($#r1_alg_1 :::"is_homomorphism"::: ) (Set (Var "A")) "," (Set (Var "B"))) & (Bool (Set (Var "h2")) ($#r1_alg_1 :::"is_homomorphism"::: ) (Set (Var "A")) "," (Set (Var "B"))) & (Bool (Set (Set (Var "h1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "G"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "h2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Var "h1")) ($#r2_funct_2 :::"="::: ) (Set (Var "h2"))))))) ; theorem :: AOFA_000:36 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o1")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o2")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o1")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o2")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p2"))))) "holds" (Bool "(" (Bool (Set (Var "o1")) ($#r1_hidden :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) ")" )))) ; theorem :: AOFA_000:37 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) (Set (Var "A"))) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "o1")))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "o2")))) & (Bool (Set (Set (Var "o1")) ($#k1_funct_1 :::"."::: ) (Set (Var "p1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "o2")) ($#k1_funct_1 :::"."::: ) (Set (Var "p2"))))) "holds" (Bool "(" (Bool (Set (Var "o1")) ($#r2_relset_1 :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) ")" )))) ; theorem :: AOFA_000:38 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "o")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" )))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")))))))) ; theorem :: AOFA_000:39 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "G")) "being" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "o")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) "st" (Bool (Bool "(" "for" (Set (Var "o9")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o9")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o9")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "G")))) "holds" (Bool (Set (Var "o9")) ($#r1_hidden :::"<>"::: ) (Set (Var "o")))) ")" )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" )))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "G")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))))))) ; theorem :: AOFA_000:40 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "o")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" )))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set (Var "o")) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k11_aofa_000 :::"Generators"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k11_aofa_000 :::"Generators"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))))))) ; begin definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) ; attr "S" is :::"with_empty-instruction"::: means :: AOFA_000:def 10 (Bool "(" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S"))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 1)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ")" ); attr "S" is :::"with_catenation"::: means :: AOFA_000:def 11 (Bool "(" (Bool (Num 2) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S"))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 2)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ")" ); attr "S" is :::"with_if-instruction"::: means :: AOFA_000:def 12 (Bool "(" (Bool (Num 3) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S"))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 3)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 3) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ")" ); attr "S" is :::"with_while-instruction"::: means :: AOFA_000:def 13 (Bool "(" (Bool (Num 4) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S"))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 4)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ")" ); attr "S" is :::"associative"::: means :: AOFA_000:def 14 (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 2)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#v1_aofa_000 :::"associative"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")); end; :: deftheorem defines :::"with_empty-instruction"::: AOFA_000:def 10 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_aofa_000 :::"with_empty-instruction"::: ) ) "iff" (Bool "(" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 1)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" ) ")" )); :: deftheorem defines :::"with_catenation"::: AOFA_000:def 11 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v4_aofa_000 :::"with_catenation"::: ) ) "iff" (Bool "(" (Bool (Num 2) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 2)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" ) ")" )); :: deftheorem defines :::"with_if-instruction"::: AOFA_000:def 12 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v5_aofa_000 :::"with_if-instruction"::: ) ) "iff" (Bool "(" (Bool (Num 3) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 3)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 3) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" ) ")" )); :: deftheorem defines :::"with_while-instruction"::: AOFA_000:def 13 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v6_aofa_000 :::"with_while-instruction"::: ) ) "iff" (Bool "(" (Bool (Num 4) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))))) & (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 4)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" ) ")" )); :: deftheorem defines :::"associative"::: AOFA_000:def 14 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v7_aofa_000 :::"associative"::: ) ) "iff" (Bool (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 2)) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#v1_aofa_000 :::"associative"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" )); definitionlet "S" be ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; attr "S" is :::"unital"::: means :: AOFA_000:def 15 (Bool "ex" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ($#k1_funct_1 :::"."::: ) (Num 2))) & (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "S") ")" ) ")" ")" ) "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) (Set (Var "f"))) ")" )); end; :: deftheorem defines :::"unital"::: AOFA_000:def 15 : (Bool "for" (Set (Var "S")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v8_aofa_000 :::"unital"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#k1_funct_1 :::"."::: ) (Num 2))) & (Bool (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ")" ) ")" ")" ) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) (Set (Var "f"))) ")" )) ")" )); theorem :: AOFA_000:41 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "c")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#v1_aofa_000 :::"associative"::: ) ($#v2_aofa_000 :::"unital"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set (Var "X")) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r1_aofa_000 :::"is_a_unity_wrt"::: ) (Set (Var "c")))) "holds" (Bool "for" (Set (Var "i")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 3) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set (Var "X")) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set (Var "X")) (Bool "for" (Set (Var "w")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_margrel1 :::"homogeneous"::: ) ($#v3_margrel1 :::"quasi_total"::: ) (Num 2) ($#v8_comput_1 :::"-ary"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" (Set (Var "X")) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set (Var "X")) (Bool "ex" (Set (Var "S")) "being" ($#v1_unialg_1 :::"strict"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "X")) ")" ) ($#k1_aofa_000 :::"-->"::: ) (Set (Var "x")) ")" ) "," (Set (Var "c")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "i")) "," (Set (Var "w")) ($#k2_finseq_4 :::"*>"::: ) ))) & (Bool (Set (Var "S")) "is" ($#v3_aofa_000 :::"with_empty-instruction"::: ) ) & (Bool (Set (Var "S")) "is" ($#v4_aofa_000 :::"with_catenation"::: ) ) & (Bool (Set (Var "S")) "is" ($#v8_aofa_000 :::"unital"::: ) ) & (Bool (Set (Var "S")) "is" ($#v7_aofa_000 :::"associative"::: ) ) & (Bool (Set (Var "S")) "is" ($#v5_aofa_000 :::"with_if-instruction"::: ) ) & (Bool (Set (Var "S")) "is" ($#v6_aofa_000 :::"with_while-instruction"::: ) ) & (Bool (Set (Var "S")) "is" ($#v3_unialg_1 :::"quasi_total"::: ) ) & (Bool (Set (Var "S")) "is" ($#v2_unialg_1 :::"partial"::: ) ) ")" ))))))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_unialg_1 :::"strict"::: ) ($#v2_unialg_1 :::"partial"::: ) ($#v3_unialg_1 :::"quasi_total"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#v7_aofa_000 :::"associative"::: ) ($#v8_aofa_000 :::"unital"::: ) for ($#l1_unialg_1 :::"UAStr"::: ) ; end; definitionmode preIfWhileAlgebra is ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::); end; definitionlet "A" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_unialg_1 :::"UAStr"::: ) ; mode Algorithm of "A" is ($#m1_subset_1 :::"Element":::) "of" "A"; end; theorem :: AOFA_000:42 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; func :::"EmptyIns"::: "A" -> ($#m1_subset_1 :::"Algorithm":::) "of" "A" equals :: AOFA_000:def 16 (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"EmptyIns"::: AOFA_000:def 16 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))); theorem :: AOFA_000:43 (Bool "for" (Set (Var "A")) "being" ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "o")) "being" ($#m5_margrel1 :::"Element"::: ) "of" (Set ($#k1_unialg_2 :::"Operations"::: ) (Set (Var "A"))) "st" (Bool (Bool (Set (Var "o")) ($#r2_relset_1 :::"="::: ) (Set ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 1) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ))) "holds" (Bool "(" (Bool (Set ($#k19_margrel1 :::"arity"::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "o")))) ")" ))) ; theorem :: AOFA_000:44 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 2) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A")))))) ; definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; let "I1", "I2" be ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Const "A")); func "I1" :::"\;"::: "I2" -> ($#m1_subset_1 :::"Algorithm":::) "of" "A" equals :: AOFA_000:def 17 (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 2) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k2_finseq_4 :::"<*"::: ) "I1" "," "I2" ($#k2_finseq_4 :::"*>"::: ) )); end; :: deftheorem defines :::"\;"::: AOFA_000:def 17 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 2) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "I1")) "," (Set (Var "I2")) ($#k2_finseq_4 :::"*>"::: ) ))))); theorem :: AOFA_000:45 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v8_aofa_000 :::"unital"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))) & (Bool (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" ))) ; theorem :: AOFA_000:46 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v7_aofa_000 :::"associative"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "I3")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "I2")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I3")) ")" ))))) ; theorem :: AOFA_000:47 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 3) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A")))))) ; definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; let "C", "I1", "I2" be ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Const "A")); func :::"if-then-else"::: "(" "C" "," "I1" "," "I2" ")" -> ($#m1_subset_1 :::"Algorithm":::) "of" "A" equals :: AOFA_000:def 18 (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 3) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k3_finseq_4 :::"<*"::: ) "C" "," "I1" "," "I2" ($#k3_finseq_4 :::"*>"::: ) )); end; :: deftheorem defines :::"if-then-else"::: AOFA_000:def 18 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 3) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ($#k3_finseq_4 :::"*>"::: ) ))))); definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; let "C", "I" be ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Const "A")); func :::"if-then"::: "(" "C" "," "I" ")" -> ($#m1_subset_1 :::"Algorithm":::) "of" "A" equals :: AOFA_000:def 19 (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" "C" "," "I" "," (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) "A" ")" ) ")" ); end; :: deftheorem defines :::"if-then"::: AOFA_000:def 19 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k15_aofa_000 :::"if-then"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" ) ")" )))); theorem :: AOFA_000:48 (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 4) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A")))))) ; definitionlet "A" be ($#v4_unialg_1 :::"non-empty"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; let "C", "I" be ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Const "A")); func :::"while"::: "(" "C" "," "I" ")" -> ($#m1_subset_1 :::"Algorithm":::) "of" "A" equals :: AOFA_000:def 20 (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 4) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k2_finseq_4 :::"<*"::: ) "C" "," "I" ($#k2_finseq_4 :::"*>"::: ) )); end; :: deftheorem defines :::"while"::: AOFA_000:def 20 : (Bool "for" (Set (Var "A")) "being" ($#v4_unialg_1 :::"non-empty"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#l1_unialg_1 :::"UAStr"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 4) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "C")) "," (Set (Var "I")) ($#k2_finseq_4 :::"*>"::: ) ))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I0", "C", "I", "J" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); func :::"for-do"::: "(" "I0" "," "C" "," "J" "," "I" ")" -> ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_000:def 21 (Set "I0" ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" "C" "," (Set "(" "I" ($#k13_aofa_000 :::"\;"::: ) "J" ")" ) ")" ")" )); end; :: deftheorem defines :::"for-do"::: AOFA_000:def 21 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I0")) "," (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k17_aofa_000 :::"for-do"::: ) "(" (Set (Var "I0")) "," (Set (Var "C")) "," (Set (Var "J")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "I0")) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J")) ")" ) ")" ")" ))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); func :::"ElementaryInstructions"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" "A" equals :: AOFA_000:def 22 (Set (Set "(" (Set "(" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) "A" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 3) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 4) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" "A") ")" ) ")" ")" ) "," "A" ")" ")" ) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) "{" (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" ) where I1, I2 "is" ($#m1_subset_1 :::"Algorithm":::) "of" "A" : (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) ")" ) "}" ); end; :: deftheorem defines :::"ElementaryInstructions"::: AOFA_000:def 22 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 3) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k2_pua2mss1 :::"Den"::: ) "(" (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Num 4) "," (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set "the" ($#u1_unialg_1 :::"charact"::: ) "of" (Set (Var "A"))) ")" ) ")" ")" ) "," (Set (Var "A")) ")" ")" ) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) "{" (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" ) where I1, I2 "is" ($#m1_subset_1 :::"Algorithm":::) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) ")" ) "}" ))); theorem :: AOFA_000:49 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r2_hidden :::"nin"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))))) ; theorem :: AOFA_000:50 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))))) "holds" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r2_hidden :::"nin"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))))) ; theorem :: AOFA_000:51 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r2_hidden :::"nin"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))))) ; theorem :: AOFA_000:52 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"nin"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))))) ; theorem :: AOFA_000:53 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" "not" (Bool (Set (Var "I")) ($#r2_hidden :::"nin"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))) "or" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) ")" )) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ))) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "J")) ")" ))) ")" ))) ; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); attr "A" is :::"infinite"::: means :: AOFA_000:def 23 (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A") "is" ($#v1_finset_1 :::"infinite"::: ) ); attr "A" is :::"degenerated"::: means :: AOFA_000:def 24 (Bool "(" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool "(" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Var "I2"))) ")" ) "or" (Bool "(" (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Var "I1"))) ")" ) "or" (Bool "(" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) "or" (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) ")" ) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) ")" ) ")" )) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A"))) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A"))) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) ")" )) ")" )) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A")) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "J")) ")" )) ")" )) "or" (Bool "ex" (Set (Var "C1")) "," (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C2")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C1")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C2")) "," (Set (Var "J")) ")" ))) ")" ); attr "A" is :::"well_founded"::: means :: AOFA_000:def 25 (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A") "is" ($#m1_freealg :::"GeneratorSet"::: ) "of" "A"); end; :: deftheorem defines :::"infinite"::: AOFA_000:def 23 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v9_aofa_000 :::"infinite"::: ) ) "iff" (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))) "is" ($#v1_finset_1 :::"infinite"::: ) ) ")" )); :: deftheorem defines :::"degenerated"::: AOFA_000:def 24 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v10_aofa_000 :::"degenerated"::: ) ) "iff" (Bool "(" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Var "I2"))) ")" ) "or" (Bool "(" (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Var "I1"))) ")" ) "or" (Bool "(" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) "or" (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) ")" ) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) ")" ) ")" )) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))))) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))))) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) ")" )) ")" )) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"<>"::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "J")) ")" )) ")" )) "or" (Bool "ex" (Set (Var "C1")) "," (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C2")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C1")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C2")) "," (Set (Var "J")) ")" ))) ")" ) ")" )); :: deftheorem defines :::"well_founded"::: AOFA_000:def 25 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v11_aofa_000 :::"well_founded"::: ) ) "iff" (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))) "is" ($#m1_freealg :::"GeneratorSet"::: ) "of" (Set (Var "A"))) ")" )); definitionfunc :::"ECIW-signature"::: -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AOFA_000:def 26 (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 2) ($#k2_finseq_4 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) ($#k2_finseq_4 :::"*>"::: ) )); end; :: deftheorem defines :::"ECIW-signature"::: AOFA_000:def 26 : (Bool (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 2) ($#k2_finseq_4 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) ($#k2_finseq_4 :::"*>"::: ) ))); theorem :: AOFA_000:54 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) )) ($#r1_hidden :::"="::: ) (Num 4)) & (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 4))) & (Bool (Set (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 2)) ")" ) ; definitionlet "A" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_unialg_1 :::"partial"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) ; attr "A" is :::"ECIW-strict"::: means :: AOFA_000:def 27 (Bool (Set ($#k1_unialg_1 :::"signature"::: ) "A") ($#r1_hidden :::"="::: ) (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) )); end; :: deftheorem defines :::"ECIW-strict"::: AOFA_000:def 27 : (Bool "for" (Set (Var "A")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_unialg_1 :::"partial"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v12_aofa_000 :::"ECIW-strict"::: ) ) "iff" (Bool (Set ($#k1_unialg_1 :::"signature"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) )) ")" )); theorem :: AOFA_000:55 (Bool "for" (Set (Var "A")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_unialg_1 :::"partial"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#l1_unialg_1 :::"UAStr"::: ) "st" (Bool (Bool (Set (Var "A")) "is" ($#v12_aofa_000 :::"ECIW-strict"::: ) )) "holds" (Bool "for" (Set (Var "o")) "being" ($#m2_subset_1 :::"OperSymbol":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "o")) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Var "o")) ($#r1_hidden :::"="::: ) (Num 2)) "or" (Bool (Set (Var "o")) ($#r1_hidden :::"="::: ) (Num 3)) "or" (Bool (Set (Var "o")) ($#r1_hidden :::"="::: ) (Num 4)) ")" ))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," "X" ")" ) -> ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ; end; theorem :: AOFA_000:56 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "holds" (Bool "(" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k2_trees_4 :::"root-tree"::: ) (Set (Var "x"))))) "or" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::)(Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 4))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) ($#k1_recdef_1 :::"."::: ) (Set (Var "n")))) ")" ))) ")" ))) ; theorem :: AOFA_000:57 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: AOFA_000:58 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Num 1) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: AOFA_000:59 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "holds" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k6_trees_4 :::"-tree"::: ) "(" (Set (Var "I1")) "," (Set (Var "I2")) ")" )))) ; theorem :: AOFA_000:60 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Num 2) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) "holds" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "I1")) "," (Set (Var "I2")) ($#k2_finseq_4 :::"*>"::: ) ))))) ; theorem :: AOFA_000:61 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set (Var "I1"))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set (Var "I2"))) ")" ))) ; theorem :: AOFA_000:62 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "J1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J2"))))) "holds" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set (Var "J1"))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Var "J2"))) ")" ))) ; theorem :: AOFA_000:63 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "holds" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 3) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ($#k3_finseq_4 :::"*>"::: ) ))))) ; theorem :: AOFA_000:64 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Num 3) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) "holds" (Bool "ex" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ($#k3_finseq_4 :::"*>"::: ) ))))) ; theorem :: AOFA_000:65 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "," (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C1")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C2")) "," (Set (Var "J1")) "," (Set (Var "J2")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "C2"))) & (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set (Var "J1"))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Var "J2"))) ")" ))) ; theorem :: AOFA_000:66 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 4) ($#k6_trees_4 :::"-tree"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" )))) ; theorem :: AOFA_000:67 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Num 4) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) "holds" (Bool "ex" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "C")) "," (Set (Var "I")) ($#k2_finseq_4 :::"*>"::: ) ))))) ; theorem :: AOFA_000:68 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" )))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k4_trees_4 :::"-tree"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))))) ; theorem :: AOFA_000:69 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "x")) ($#k4_trees_4 :::"-tree"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; theorem :: AOFA_000:70 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set "(" ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_freealg :::"FreeGenSetNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" )) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k12_freealg :::"FreeGenSetNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set (Var "X")) ")" ")" ))) ")" )) ; registration cluster ($#v1_finset_1 :::"infinite"::: ) ($#v1_freealg :::"disjoint_with_NAT"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "X" be ($#v1_finset_1 :::"infinite"::: ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," "X" ")" ) -> ($#v9_aofa_000 :::"infinite"::: ) ; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," "X" ")" ) -> ($#v12_aofa_000 :::"ECIW-strict"::: ) ; end; theorem :: AOFA_000:71 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool (Set ($#k10_aofa_000 :::"Generators"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))))) ; theorem :: AOFA_000:72 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) )) "holds" (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" )) & (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) ")" )) ")" ))) ; theorem :: AOFA_000:73 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) )) "holds" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set (Var "I1"))) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set (Var "I2"))) & "(" (Bool (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "J1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J2"))))) "implies" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set (Var "J1"))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Var "J2"))) ")" ) ")" & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "J1")) "," (Set (Var "J2")) ")" )) & (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r1_hidden :::"<>"::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "J1")) ")" )) ")" ))) ; theorem :: AOFA_000:74 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) )) "holds" (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "," (Set (Var "D")) "," (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "C"))) & (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "I1"))) & (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "I2"))) & (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "D")) "," (Set (Var "J1")) ")" )) & "(" (Bool (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "D")) "," (Set (Var "J1")) "," (Set (Var "J2")) ")" ))) "implies" (Bool "(" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set (Var "J1"))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Var "J2"))) ")" ) ")" ")" ))) ; theorem :: AOFA_000:75 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) )) "holds" (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "D")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "C"))) & (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "I"))) & "(" (Bool (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "D")) "," (Set (Var "J")) ")" ))) "implies" (Bool "(" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Var "J"))) ")" ) ")" ")" ))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_unialg_1 :::"partial"::: ) ($#v3_unialg_1 :::"quasi_total"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_freealg :::"free"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) -> ($#~v10_aofa_000 "non" ($#v10_aofa_000 :::"degenerated"::: ) ) ($#v11_aofa_000 :::"well_founded"::: ) for ($#l1_unialg_1 :::"UAStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_unialg_1 :::"strict"::: ) ($#v2_unialg_1 :::"partial"::: ) ($#v3_unialg_1 :::"quasi_total"::: ) ($#v4_unialg_1 :::"non-empty"::: ) ($#v3_freealg :::"free"::: ) ($#v3_aofa_000 :::"with_empty-instruction"::: ) ($#v4_aofa_000 :::"with_catenation"::: ) ($#v5_aofa_000 :::"with_if-instruction"::: ) ($#v6_aofa_000 :::"with_while-instruction"::: ) ($#v9_aofa_000 :::"infinite"::: ) ($#~v10_aofa_000 "non" ($#v10_aofa_000 :::"degenerated"::: ) ) ($#v11_aofa_000 :::"well_founded"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) for ($#l1_unialg_1 :::"UAStr"::: ) ; end; definitionmode IfWhileAlgebra is ($#~v10_aofa_000 "non" ($#v10_aofa_000 :::"degenerated"::: ) ) ($#v11_aofa_000 :::"well_founded"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); end; registrationlet "A" be ($#v9_aofa_000 :::"infinite"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); cluster (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A") -> ($#v1_finset_1 :::"infinite"::: ) ; end; theorem :: AOFA_000:76 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool "(" "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I1")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I2")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))) "holds" (Bool "(" (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) ")" )))) ; theorem :: AOFA_000:77 (Bool "for" (Set (Var "A")) "being" ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")))) "or" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" )) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" )) & (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" )) "or" (Bool "ex" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" )) & (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) & (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))) ")" )) ")" )))) ; theorem :: AOFA_000:78 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"Universal_Algebra":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k6_unialg_2 :::"Constants"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_aofa_000 :::"|^"::: ) (Num 1))))) ; theorem :: AOFA_000:79 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v11_aofa_000 :::"well_founded"::: ) ) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n")))))) ")" )) ; scheme :: AOFA_000:sch 2 StructInd{ F1() -> ($#v11_aofa_000 :::"well_founded"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool P1[(Set F2 "(" ")" )]) provided (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set F1 "(" ")" )))) "holds" (Bool P1[(Set (Var "I"))])) and (Bool P1[(Set ($#k12_aofa_000 :::"EmptyIns"::: ) (Set F1 "(" ")" ))]) and (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "I1"))]) & (Bool P1[(Set (Var "I2"))])) "holds" (Bool P1[(Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")))])) and (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "C"))]) & (Bool P1[(Set (Var "I1"))]) & (Bool P1[(Set (Var "I2"))])) "holds" (Bool P1[(Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" )])) and (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "C"))]) & (Bool P1[(Set (Var "I"))])) "holds" (Bool P1[(Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" )])) proof end; begin definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "S")); attr "f" is :::"complying_with_empty-instruction"::: means :: AOFA_000:def 28 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) "A" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "s")))); attr "f" is :::"complying_with_catenation"::: means :: AOFA_000:def 29 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I1")) ")" ")" ) "," (Set (Var "I2")) ")" )))); end; :: deftheorem defines :::"complying_with_empty-instruction"::: AOFA_000:def 28 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "s")))) ")" )))); :: deftheorem defines :::"complying_with_catenation"::: AOFA_000:def 29 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v14_aofa_000 :::"complying_with_catenation"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I1")) ")" ")" ) "," (Set (Var "I2")) ")" )))) ")" )))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "S")); pred "f" :::"complies_with_if_wrt"::: "T" means :: AOFA_000:def 30 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "holds" (Bool "(" "(" (Bool (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) "T")) "implies" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I1")) ")" )) ")" & "(" (Bool (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) "T")) "implies" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I2")) ")" )) ")" ")" ))); pred "f" :::"complies_with_while_wrt"::: "T" means :: AOFA_000:def 31 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "holds" (Bool "(" "(" (Bool (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) "T")) "implies" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ")" ")" ) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" )) ")" & "(" (Bool (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) "T")) "implies" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) ")" ")" ))); end; :: deftheorem defines :::"complies_with_if_wrt"::: AOFA_000:def 30 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_aofa_000 :::"complies_with_if_wrt"::: ) (Set (Var "T"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "implies" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I1")) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "implies" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I2")) ")" )) ")" ")" ))) ")" ))))); :: deftheorem defines :::"complies_with_while_wrt"::: AOFA_000:def 31 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_aofa_000 :::"complies_with_while_wrt"::: ) (Set (Var "T"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "implies" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ")" ")" ) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "implies" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) ")" ")" ))) ")" ))))); theorem :: AOFA_000:80 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ) & (Bool (Set (Var "f")) ($#r2_aofa_000 :::"complies_with_if_wrt"::: ) (Set (Var "T")))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k15_aofa_000 :::"if-then"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )))))))) ; theorem :: AOFA_000:81 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set ($#k6_trees_3 :::"pr1"::: ) "(" (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ")" ) "is" ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ) & (Bool (Set ($#k6_trees_3 :::"pr1"::: ) "(" (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ")" ) "is" ($#v14_aofa_000 :::"complying_with_catenation"::: ) ) & (Bool (Set ($#k6_trees_3 :::"pr1"::: ) "(" (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ")" ) ($#r2_aofa_000 :::"complies_with_if_wrt"::: ) (Set (Var "T"))) & (Bool (Set ($#k6_trees_3 :::"pr1"::: ) "(" (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ")" ) ($#r3_aofa_000 :::"complies_with_while_wrt"::: ) (Set (Var "T"))) ")" )))) ; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); mode :::"ExecutionFunction"::: "of" "A" "," "S" "," "T" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "S" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k2_zfmisc_1 :::":]"::: ) ) "," "S" means :: AOFA_000:def 32 (Bool "(" (Bool it "is" ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ) & (Bool it "is" ($#v14_aofa_000 :::"complying_with_catenation"::: ) ) & (Bool it ($#r2_aofa_000 :::"complies_with_if_wrt"::: ) "T") & (Bool it ($#r3_aofa_000 :::"complies_with_while_wrt"::: ) "T") ")" ); end; :: deftheorem defines :::"ExecutionFunction"::: AOFA_000:def 32 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T"))) "iff" (Bool "(" (Bool (Set (Var "b4")) "is" ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ) & (Bool (Set (Var "b4")) "is" ($#v14_aofa_000 :::"complying_with_catenation"::: ) ) & (Bool (Set (Var "b4")) ($#r2_aofa_000 :::"complies_with_if_wrt"::: ) (Set (Var "T"))) & (Bool (Set (Var "b4")) ($#r3_aofa_000 :::"complies_with_while_wrt"::: ) (Set (Var "T"))) ")" ) ")" ))))); registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); cluster -> ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ($#v14_aofa_000 :::"complying_with_catenation"::: ) for ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" "A" "," "S" "," "T"; end; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set (Const "S")) "," (Set (Const "T")); pred "f" :::"iteration_terminates_for"::: "I" "," "s" means :: AOFA_000:def 33 (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "s") & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) "T") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) "T") & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," "I" ")" )) ")" ) ")" ) ")" )); end; :: deftheorem defines :::"iteration_terminates_for"::: AOFA_000:def 33 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s"))) "iff" (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set (Var "I")) ")" )) ")" ) ")" ) ")" )) ")" ))))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set (Const "S")) "," (Set (Const "T")); func :::"iteration-degree"::: "(" "I" "," "s" "," "f" ")" -> ($#m1_subset_1 :::"R_eal":::) means :: AOFA_000:def 34 (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "s") & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) "T") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) "T") & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," "I" ")" )) ")" ) ")" ) ")" )) if (Bool "f" ($#r4_aofa_000 :::"iteration_terminates_for"::: ) "I" "," "s") otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_supinf_1 :::"+infty"::: ) )); end; :: deftheorem defines :::"iteration-degree"::: AOFA_000:def 34 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "b7")) "being" ($#m1_subset_1 :::"R_eal":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s")))) "implies" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" )) "iff" (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set (Var "I")) ")" )) ")" ) ")" ) ")" )) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s"))))) "implies" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" )) "iff" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k1_supinf_1 :::"+infty"::: ) )) ")" ) ")" ")" )))))))); theorem :: AOFA_000:82 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s"))) "iff" (Bool (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_supinf_1 :::"+infty"::: ) )) ")" ))))))) ; theorem :: AOFA_000:83 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s"))) & (Bool (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))))))) ; theorem :: AOFA_000:84 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s")))) "implies" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" )) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ))) "implies" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s"))) ")" & (Bool (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k8_mesfunc1 :::"1."::: ) ) ($#k3_supinf_2 :::"+"::: ) (Set "(" ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) "," (Set (Var "f")) ")" ")" ))) ")" ))))))) ; theorem :: AOFA_000:85 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set ($#k20_aofa_000 :::"iteration-degree"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) "," (Set (Var "f")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))))))) ; scheme :: AOFA_000:sch 3 Termination{ F1() -> ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F3() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F4() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ), F5() -> ($#m1_subset_1 :::"Subset":::) "of" (Set F3 "(" ")" ), F6() -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set F3 "(" ")" ) "," (Set F5 "(" ")" ), F7( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool (Set F6 "(" ")" ) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set F2 "(" ")" ) "," (Set F4 "(" ")" )) provided (Bool "(" (Bool (Set F4 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" )) "iff" (Bool P1[(Set F4 "(" ")" )]) ")" ) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" "(" (Bool (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )])) "implies" (Bool (Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" )) ")" & "(" (Bool (Bool (Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" ))) "implies" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )]) ")" & (Bool (Set F7 "(" (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set F7 "(" (Set (Var "s")) ")" )) ")" )) proof end; scheme :: AOFA_000:sch 4 Termination2{ F1() -> ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F3() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F4() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ), F5() -> ($#m1_subset_1 :::"Subset":::) "of" (Set F3 "(" ")" ), F6() -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set F3 "(" ")" ) "," (Set F5 "(" ")" ), F7( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) ] } : (Bool (Set F6 "(" ")" ) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set F2 "(" ")" ) "," (Set F4 "(" ")" )) provided (Bool P1[(Set F4 "(" ")" )]) and (Bool "(" (Bool (Set F4 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" )) "iff" (Bool P2[(Set F4 "(" ")" )]) ")" ) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "s"))]) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" )) & (Bool P2[(Set (Var "s"))])) "holds" (Bool "(" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )]) & "(" (Bool (Bool P2[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )])) "implies" (Bool (Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" )) ")" & "(" (Bool (Bool (Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F5 "(" ")" ))) "implies" (Bool P2[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )]) ")" & (Bool (Set F7 "(" (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set F7 "(" (Set (Var "s")) ")" )) ")" )) proof end; theorem :: AOFA_000:86 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )))))))))) ; theorem :: AOFA_000:87 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Var "I")) "," (Set (Var "s")))) "iff" (Bool (Set (Set "(" (Set "(" ($#k12_funct_5 :::"curry'"::: ) (Set (Var "f")) ")" ) ($#k10_funct_5 :::"."::: ) (Set (Var "I")) ")" ) ($#k5_aofa_000 :::"orbit"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Var "T"))) ")" ))))))) ; scheme :: AOFA_000:sch 5 InvariantSch{ F1() -> ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F3() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F4() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F5() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ), F6() -> ($#m1_subset_1 :::"Subset":::) "of" (Set F4 "(" ")" ), F7() -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set F4 "(" ")" ) "," (Set F6 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) ] } : (Bool "(" (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F5 "(" ")" ) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ")" ) ")" )]) & (Bool P2[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F5 "(" ")" ) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ")" ) ")" )]) ")" ) provided (Bool P1[(Set F5 "(" ")" )]) and (Bool (Set F7 "(" ")" ) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set F3 "(" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set F2 "(" ")" )) "," (Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F5 "(" ")" ) "," (Set F2 "(" ")" ) ")" )) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "s"))]) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F6 "(" ")" )) & (Bool P2[(Set (Var "s"))])) "holds" (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F3 "(" ")" ) ")" )])) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )]) & "(" (Bool (Bool (Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F6 "(" ")" ))) "implies" (Bool P2[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )]) ")" & "(" (Bool (Bool P2[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )])) "implies" (Bool (Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F6 "(" ")" )) ")" ")" )) proof end; scheme :: AOFA_000:sch 6 coInvariantSch{ F1() -> ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F3() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F4() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F5() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ), F6() -> ($#m1_subset_1 :::"Subset":::) "of" (Set F4 "(" ")" ), F7() -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set F4 "(" ")" ) "," (Set F6 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool P1[(Set F5 "(" ")" )]) provided (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F5 "(" ")" ) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ")" ) ")" )]) and (Bool (Set F7 "(" ")" ) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set F3 "(" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set F2 "(" ")" )) "," (Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F5 "(" ")" ) "," (Set F2 "(" ")" ) ")" )) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ) "st" (Bool (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ")" ) "," (Set F3 "(" ")" ) ")" )]) & (Bool (Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set F6 "(" ")" ))) "holds" (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )])) and (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F4 "(" ")" ) "st" (Bool (Bool P1[(Set (Set F7 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" )])) "holds" (Bool P1[(Set (Var "s"))])) proof end; theorem :: AOFA_000:88 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "I1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) & (Bool (Set (Var "I2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) ")" ))))) ; theorem :: AOFA_000:89 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) & (Bool (Set (Var "I1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) & (Bool (Set (Var "I2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) ")" ))))) ; theorem :: AOFA_000:90 (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "n"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) & (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k9_aofa_000 :::"|^"::: ) (Set (Var "i")))) ")" ))))) ; begin scheme :: AOFA_000:sch 7 IndDef{ F1() -> ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ), F4( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) , F5( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ), F6( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ), F7( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F1 "(" ")" )) "," (Set F2 "(" ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "I")) ")" )) ")" ) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set F1 "(" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set F3 "(" ")" )) & (Bool "(" "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "I1")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "I2")) ")" )) ($#r1_hidden :::"="::: ) (Set F5 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I1")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I2")) ")" ) ")" )) ")" ) & (Bool "(" "for" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I1")) "," (Set (Var "I2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set F7 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "C")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I1")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I2")) ")" ) ")" )) ")" ) & (Bool "(" "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set F6 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "C")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ")" )) ")" ) ")" )) provided (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set F4 "(" (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) proof end; theorem :: AOFA_000:91 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) (Bool "for" (Set (Var "s0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "ex" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Bool "not" (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "s0")))) ")" ) ")" ))))))) ; theorem :: AOFA_000:92 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "S")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "S")) "," (Set (Var "S")) ")" ")" ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "S")) "," (Set (Var "S")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "h")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "S")) "," (Set (Var "S")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "F")) ($#k10_funct_5 :::"."::: ) (Set (Var "h")) ")" ) ($#k4_aofa_000 :::"*"::: ) (Set (Var "h"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "F")) ($#k10_funct_5 :::"."::: ) (Set (Var "h")))) ")" )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Bool "not" (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F")) ($#k10_funct_5 :::"."::: ) (Set "(" (Set "(" ($#k12_funct_5 :::"curry'"::: ) (Set (Var "f")) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" )))) ")" ) ")" ))))))) ; theorem :: AOFA_000:93 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Bool "not" (Set (Var "f1")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f1")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ")" ))) ")" )) "holds" (Bool (Set (Var "f1")) ($#r8_binop_1 :::"="::: ) (Set (Var "f2"))))))) ; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set (Const "S")) "," (Set (Const "T")); func :::"TerminatingPrograms"::: "(" "A" "," "S" "," "T" "," "f" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "S" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k2_zfmisc_1 :::":]"::: ) ) means :: AOFA_000:def 35 (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) "S" "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) it) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) "S" "," (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) "A" ")" ) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) it) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "holds" (Bool "(" "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) "," (Set (Var "J")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it)) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) ")" & "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) "T")) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) ")" & "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "J")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) "T")) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) ")" & "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) & (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) "T") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) "T") & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) 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(Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" & "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "J")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" & "(" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ")" )) ")" ) ")" ) ")" ))) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" ")" )) ")" )) "holds" (Bool (Set (Var "b5")) ($#r1_relset_1 :::"c="::: ) (Set (Var "P"))) ")" ) ")" ) ")" )))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); attr "I" is :::"absolutely-terminating"::: means :: AOFA_000:def 36 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" "A" "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," "I" ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" "A" "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))); end; :: deftheorem defines :::"absolutely-terminating"::: AOFA_000:def 36 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v15_aofa_000 :::"absolutely-terminating"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))) ")" ))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set (Const "S")) "," (Set (Const "T")); pred "I" :::"is_terminating_wrt"::: "f" means :: AOFA_000:def 37 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," "I" ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" "A" "," "S" "," "T" "," "f" ")" ))); end; :: deftheorem defines :::"is_terminating_wrt"::: AOFA_000:def 37 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))) ")" )))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set (Const "S")) "," (Set (Const "T")); let "Z" be ($#m1_hidden :::"set"::: ) ; pred "I" :::"is_terminating_wrt"::: "f" "," "Z" means :: AOFA_000:def 38 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," "I" ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" "A" "," "S" "," "T" "," "f" ")" ))); pred "Z" :::"is_invariant_wrt"::: "I" "," "f" means :: AOFA_000:def 39 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," "I" ")" ) ($#r2_hidden :::"in"::: ) "Z")); end; :: deftheorem defines :::"is_terminating_wrt"::: AOFA_000:def 38 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "Z"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))) ")" ))))))); :: deftheorem defines :::"is_invariant_wrt"::: AOFA_000:def 39 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "Z")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "f"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) ")" ))))))); theorem :: AOFA_000:94 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))))) ; theorem :: AOFA_000:95 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "I")) "is" ($#v15_aofa_000 :::"absolutely-terminating"::: ) ))) ; theorem :: AOFA_000:96 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k12_aofa_000 :::"EmptyIns"::: ) (Set (Var "A")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))))))) ; registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); cluster (Set ($#k12_aofa_000 :::"EmptyIns"::: ) "A") -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ; end; registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); cluster ($#v15_aofa_000 :::"absolutely-terminating"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A"); end; theorem :: AOFA_000:97 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) "," (Set (Var "J")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) ")" ))))))) ; registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I", "J" be ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); cluster (Set "I" ($#k13_aofa_000 :::"\;"::: ) "J") -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ; end; theorem :: AOFA_000:98 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) & "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"nin"::: ) (Set (Var "T")))) "implies" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "J")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) ")" ")" ))))))) ; registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "C", "I", "J" be ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); cluster (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" "C" "," "I" "," "J" ")" ) -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ; end; registrationlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "C", "I" be ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); cluster (Set ($#k15_aofa_000 :::"if-then"::: ) "(" "C" "," "I" ")" ) -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ; end; theorem :: AOFA_000:99 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "C")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) & (Bool "ex" (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )) ($#r2_hidden :::"nin"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) & (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C")) ")" ) ")" )) ")" ) ")" ) ")" )) ")" ))))))) ; theorem :: AOFA_000:100 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_freealg :::"free"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )) & (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ")" ) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))))) ; theorem :: AOFA_000:101 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))))) ; theorem :: AOFA_000:102 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))) "holds" (Bool (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f1")) ")" ) ($#r2_relset_1 :::"="::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f2")) ")" )))))) ; theorem :: AOFA_000:103 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v3_freealg :::"free"::: ) ($#v12_aofa_000 :::"ECIW-strict"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "S")) "," (Set "(" ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f1")) ")" ))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" )))))))) ; theorem :: AOFA_000:104 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))))))) ; theorem :: AOFA_000:105 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) "iff" (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "S"))) ")" )))))) ; theorem :: AOFA_000:106 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))) "holds" (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))))))))) ; theorem :: AOFA_000:107 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))))))))) ; theorem :: AOFA_000:108 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "holds" (Bool (Set (Var "S")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "f")))))))) ; theorem :: AOFA_000:109 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "f"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "J")) "," (Set (Var "f")))) "holds" (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J"))) "," (Set (Var "f"))))))))) ; theorem :: AOFA_000:110 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "J")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))) "holds" (Bool (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J"))) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))))))) ; theorem :: AOFA_000:111 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "J")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "f")))) "holds" (Bool (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "J"))) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))))))))) ; theorem :: AOFA_000:112 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "J")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))))))) ; theorem :: AOFA_000:113 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "J")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "C")) "," (Set (Var "f")))) "holds" (Bool (Set ($#k14_aofa_000 :::"if-then-else"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) "," (Set (Var "J")) ")" ) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))))))))) ; theorem :: AOFA_000:114 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" )))))))) ; theorem :: AOFA_000:115 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "C")) "," (Set (Var "f"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "f"))) & (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))))))))) ; theorem :: AOFA_000:116 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "C")) "," (Set (Var "f"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" ) & (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k21_aofa_000 :::"TerminatingPrograms"::: ) "(" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "f")) ")" ))))))))) ; theorem :: AOFA_000:117 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Var "s"))) ")" )) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")))))))) ; theorem :: AOFA_000:118 (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "C")) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "C")) "," (Set (Var "f"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" ) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Var "f")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set (Var "C"))) "," (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "C")) ")" )) ")" )) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set (Var "C")) "," (Set (Var "I")) ")" ) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "f")) "," (Set (Var "P"))))))))) ;