:: AOFA_I00 semantic presentation begin theorem :: AOFA_I00:1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_hidden :::"<>"::: ) (Set (Var "a")))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Var "z"))) ")" ))) ; definitionlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "x", "f" be ($#m1_hidden :::"set"::: ) ; func "F" :::"\"::: "(" "x" "," "f" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "F" equals :: AOFA_I00:def 1 "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Element"::: ) "of" "F" : (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) "x") ($#r1_hidden :::"<>"::: ) "f") "}" ; end; :: deftheorem defines :::"\"::: AOFA_I00:def 1 : (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "F")) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "F")) : (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set (Var "f"))) "}" ))); theorem :: AOFA_I00:2 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) "iff" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" )))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "Y", "Z" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set (Const "Y")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "Z")); mode :::"Variable"::: "of" "f" -> ($#m1_subset_1 :::"Element"::: ) "of" "X" means :: AOFA_I00:def 2 (Bool verum); end; :: deftheorem defines :::"Variable"::: AOFA_I00:def 2 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "Z")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f"))) "iff" (Bool verum) ")" )))); notationlet "f" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; synonym "f" :::"*"::: "x" for "x" :::"(#)"::: "f"; end; definitionlet "t1", "t2" be (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); func "t1" :::"div"::: "t2" -> (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 3 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t1" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t2" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" "t2" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ))) ")" ) ")" ); func "t1" :::"mod"::: "t2" -> (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 4 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t1" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t2" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" "t2" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ))) ")" ) ")" ); func :::"leq"::: "(" "t1" "," "t2" ")" -> (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 5 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t1" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t2" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" "t1" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) ")" ) ")" ); func :::"gt"::: "(" "t1" "," "t2" ")" -> (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 6 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t1" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t2" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" "t1" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" ) ")" ); func :::"eq"::: "(" "t1" "," "t2" ")" -> (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 7 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t1" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "t2" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set "(" "t1" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"div"::: AOFA_I00:def 3 : (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b3")) "being" (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k2_aofa_i00 :::"div"::: ) (Set (Var "t2")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t2")) ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" (Set (Var "t2")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ))) ")" ) ")" ) ")" )); :: deftheorem defines :::"mod"::: AOFA_I00:def 4 : (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b3")) "being" (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k3_aofa_i00 :::"mod"::: ) (Set (Var "t2")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t2")) ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" (Set (Var "t2")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ))) ")" ) ")" ) ")" )); :: deftheorem defines :::"leq"::: AOFA_I00:def 5 : (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b3")) "being" (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_aofa_i00 :::"leq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t2")) ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" (Set (Var "t1")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) ")" ) ")" ) ")" )); :: deftheorem defines :::"gt"::: AOFA_I00:def 6 : (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b3")) "being" (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_aofa_i00 :::"gt"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t2")) ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" (Set (Var "t1")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" ) ")" ) ")" )); :: deftheorem defines :::"eq"::: AOFA_I00:def 7 : (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b3")) "being" (Set ($#k4_numbers :::"INT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_aofa_i00 :::"eq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "t2")) ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set "(" (Set (Var "t1")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" ) ")" ) ")" )); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ); let "x" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; :: original: :::"+"::: redefine func "f" :::"+"::: "x" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: AOFA_I00:def 8 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) "x"))); :: original: :::"-"::: redefine func "f" :::"-"::: "x" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: AOFA_I00:def 9 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "x"))); :: original: :::"*"::: redefine func "f" :::"*"::: "x" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: AOFA_I00:def 10 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) "x"))); end; :: deftheorem defines :::"+"::: AOFA_I00:def 8 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_aofa_i00 :::"+"::: ) (Set (Var "x")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "x"))))) ")" ))))); :: deftheorem defines :::"-"::: AOFA_I00:def 9 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_aofa_i00 :::"-"::: ) (Set (Var "x")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x"))))) ")" ))))); :: deftheorem defines :::"*"::: AOFA_I00:def 10 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k9_aofa_i00 :::"*"::: ) (Set (Var "x")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "x"))))) ")" ))))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "f", "g" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"div"::: redefine func "f" :::"div"::: "g" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"mod"::: redefine func "f" :::"mod"::: "g" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"leq"::: redefine func :::"leq"::: "(" "f" "," "g" ")" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"gt"::: redefine func :::"gt"::: "(" "f" "," "g" ")" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"eq"::: redefine func :::"eq"::: "(" "f" "," "g" ")" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ); end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "t1", "t2" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"-"::: redefine func "t1" :::"-"::: "t2" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: AOFA_I00:def 11 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))); :: original: :::"+"::: redefine func "t1" :::"+"::: "t2" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: AOFA_I00:def 12 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))); end; :: deftheorem defines :::"-"::: AOFA_I00:def 11 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k15_aofa_i00 :::"-"::: ) (Set (Var "t2")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))) ")" ))); :: deftheorem defines :::"+"::: AOFA_I00:def 12 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k16_aofa_i00 :::"+"::: ) (Set (Var "t2")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))) ")" ))); registrationlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "B" be ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_funct_2 :::"Funcs"::: ) "(" "A" "," "B" ")" ) -> ($#v1_finset_1 :::"infinite"::: ) ; end; definitionlet "N" be ($#m1_hidden :::"set"::: ) ; let "v", "f" be ($#m1_hidden :::"Function":::); func "v" :::"**"::: "(" "f" "," "N" ")" -> ($#m1_hidden :::"Function":::) means :: AOFA_I00:def 13 (Bool "(" (Bool "ex" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) "iff" (Bool "ex" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "v")) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h")))) ")" )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it)) "iff" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" "N" "," (Set (Var "Y")) ")" )) & (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) "f") ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "v")) ")" )) ")" ) ")" ) ")" ) ")" )) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set "v" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) "f" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"**"::: AOFA_I00:def 13 : (Bool "for" (Set (Var "N")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "f")) "," (Set (Var "b4")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k17_aofa_i00 :::"**"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" )) "iff" (Bool "(" (Bool "ex" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) "iff" (Bool "ex" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "v")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h")))) ")" )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b4")))) "iff" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "N")) "," (Set (Var "Y")) ")" )) & (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "v")))) ")" )) ")" ) ")" ) ")" ) ")" )) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))) ")" ) ")" ) ")" ))); definitionlet "X", "Y", "Z", "N" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "v" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ")" ) "," (Set (Const "Z")) ")" ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "N")); :: original: :::"**"::: redefine func "v" :::"**"::: "(" "f" "," "N" ")" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "N" "," "Y" ")" ")" ) "," "Z" ")" ); end; theorem :: AOFA_I00:3 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "f2")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))))) ; theorem :: AOFA_I00:4 (Bool "for" (Set (Var "X")) "," (Set (Var "N")) "," (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "I")) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "N")) "st" (Bool (Bool (Set (Var "c")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set "(" (Set (Var "N")) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "n")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "s")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "c")) ($#k2_funct_1 :::"""::: ) ")" ) ")" )) "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "N")) "," (Set (Var "I"))))))) ; theorem :: AOFA_I00:5 (Bool "for" (Set (Var "N")) "," (Set (Var "X")) "," (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "v2")))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "I")) ")" ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "N")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "v1")) ($#k17_aofa_i00 :::"**"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "v2")) ($#k17_aofa_i00 :::"**"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" ))) "holds" (Bool (Set (Var "v1")) ($#r1_hidden :::"="::: ) (Set (Var "v2")))))) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_card_1 :::"card"::: ) "X") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," (Set "(" ($#k1_card_1 :::"card"::: ) "X" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_card_1 :::"card"::: ) "X") ($#v4_relat_1 :::"-defined"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k1_card_1 :::"card"::: ) "X" ")" ) "," "X" ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; mode Enumeration of "X" is ($#v2_funct_1 :::"one-to-one"::: ) ($#v2_funct_2 :::"onto"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "(" ($#k1_card_1 :::"card"::: ) "X" ")" ); mode Denumeration of "X" is ($#v2_funct_1 :::"one-to-one"::: ) ($#v2_funct_2 :::"onto"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_card_1 :::"card"::: ) "X" ")" ) "," "X"; end; theorem :: AOFA_I00:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X"))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) ")" ))) ; theorem :: AOFA_I00:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Denumeration":::) "of" (Set (Var "X"))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) ")" ))) ; theorem :: AOFA_I00:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k15_funct_7 :::"+*"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k15_funct_7 :::"+*"::: ) "(" (Set (Var "y")) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ")" ) "is" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")))))) ; theorem :: AOFA_I00:9 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) "st" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: AOFA_I00:10 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Denumeration":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) ; theorem :: AOFA_I00:11 (Bool "for" (Set (Var "X")) "being" ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_numbers :::"NAT"::: ) )))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Const "X")); :: original: :::"""::: redefine func "f" :::"""::: -> ($#m1_subset_1 :::"Denumeration":::) "of" "X"; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Denumeration":::) "of" (Set (Const "X")); :: original: :::"""::: redefine func "f" :::"""::: -> ($#m1_subset_1 :::"Enumeration":::) "of" "X"; end; theorem :: AOFA_I00:12 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k3_power :::"to_power"::: ) (Set (Var "m")) ")" )))) ; theorem :: AOFA_I00:13 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "m")) ")" ))))) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; mode INT-Variable of "X" is ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," "X"; mode INT-Expression of "X" is ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ); mode INT-Array of "X" is ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_numbers :::"INT"::: ) ) "," "X"; end; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); pred "I" :::"is_assignment_wrt"::: "A" "," "X" "," "f" means :: AOFA_I00:def 14 (Bool "(" (Bool "I" ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A")) & (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" "X"(Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" "X" "st" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," "I" ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))) ")" ); end; :: deftheorem defines :::"is_assignment_wrt"::: AOFA_I00:def 14 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r1_aofa_i00 :::"is_assignment_wrt"::: ) (Set (Var "A")) "," (Set (Var "X")) "," (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))) & (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X"))(Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "st" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))) ")" ) ")" )))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Const "X")); let "t" be ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Const "X")); pred "v" "," "t" :::"form_assignment_wrt"::: "f" means :: AOFA_I00:def 15 (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" "A" "st" (Bool "(" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A")) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" "v" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ")" ) ")" )); end; :: deftheorem defines :::"form_assignment_wrt"::: AOFA_I00:def 15 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "v")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "I")) "being" ($#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")))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ")" ) ")" )) ")" ))))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); assume (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")) "st" (Bool (Set (Var "I")) ($#r1_aofa_i00 :::"is_assignment_wrt"::: ) (Set (Const "A")) "," (Set (Const "X")) "," (Set (Const "f")))) ; mode :::"INT-Variable"::: "of" "A" "," "f" -> ($#m1_subset_1 :::"INT-Variable":::) "of" "X" means :: AOFA_I00:def 16 (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" "X" "st" (Bool it "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) "f")); end; :: deftheorem defines :::"INT-Variable"::: AOFA_I00:def 16 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "st" (Bool (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set (Var "I")) ($#r1_aofa_i00 :::"is_assignment_wrt"::: ) (Set (Var "A")) "," (Set (Var "X")) "," (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "b5")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f")))) ")" )))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); assume (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")) "st" (Bool (Set (Var "I")) ($#r1_aofa_i00 :::"is_assignment_wrt"::: ) (Set (Const "A")) "," (Set (Const "X")) "," (Set (Const "f")))) ; mode :::"INT-Expression"::: "of" "A" "," "f" -> ($#m1_subset_1 :::"INT-Expression":::) "of" "X" means :: AOFA_I00:def 17 (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" "X" "st" (Bool (Set (Var "v")) "," it ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) "f")); end; :: deftheorem defines :::"INT-Expression"::: AOFA_I00:def 17 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "st" (Bool (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Set (Var "I")) ($#r1_aofa_i00 :::"is_assignment_wrt"::: ) (Set (Var "A")) "," (Set (Var "X")) "," (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "v")) "," (Set (Var "b5")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f")))) ")" )))))); definitionlet "X", "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); :: original: :::"."::: redefine func "f" :::"."::: "x" -> ($#m1_subset_1 :::"Element"::: ) "of" "Y"; end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); func :::"."::: "x" -> ($#m1_subset_1 :::"INT-Expression":::) "of" "X" means :: AOFA_I00:def 18 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) "x"))); end; :: deftheorem defines :::"."::: AOFA_I00:def 18 : (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 "b3")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k22_aofa_i00 :::"."::: ) (Set (Var "x")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))))) ")" )))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "v" be ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Const "X")); func :::"."::: "v" -> ($#m1_subset_1 :::"INT-Expression":::) "of" "X" means :: AOFA_I00:def 19 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set "(" "v" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))); end; :: deftheorem defines :::"."::: AOFA_I00:def 19 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k23_aofa_i00 :::"."::: ) (Set (Var "v")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))) ")" )))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); func :::"^"::: "x" -> ($#m1_subset_1 :::"INT-Variable":::) "of" "X" equals :: AOFA_I00:def 20 (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k8_funcop_1 :::"-->"::: ) "x"); end; :: deftheorem defines :::"^"::: AOFA_I00:def 20 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k24_aofa_i00 :::"^"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "x")))))); theorem :: AOFA_I00:14 (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")) "holds" (Bool (Set ($#k22_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r2_funct_2 :::"="::: ) (Set ($#k23_aofa_i00 :::"."::: ) (Set "(" ($#k24_aofa_i00 :::"^"::: ) (Set (Var "x")) ")" ))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func :::"."::: "(" "i" "," "X" ")" -> ($#m1_subset_1 :::"INT-Expression":::) "of" "X" equals :: AOFA_I00:def 21 (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k7_funcop_1 :::"-->"::: ) "i"); end; :: deftheorem defines :::"."::: AOFA_I00:def 21 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k25_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "i")))))); theorem :: AOFA_I00:15 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k16_aofa_i00 :::"+"::: ) (Set "(" ($#k25_aofa_i00 :::"."::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "X")) ")" ")" )) ($#r2_funct_2 :::"="::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "t")) ($#k22_valued_1 :::"(#)"::: ) (Set "(" ($#k25_aofa_i00 :::"."::: ) "(" (Num 1) "," (Set (Var "X")) ")" ")" )) ($#r2_funct_2 :::"="::: ) (Set (Var "t"))) ")" ))) ; definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); attr "f" is :::"Euclidean"::: means :: AOFA_I00:def 22 (Bool "(" (Bool "(" "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" "A" "," "f" (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" "holds" (Bool (Set (Var "v")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) "f")) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k25_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," "X" ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" "A" "," "f" "holds" (Bool (Set ($#k23_aofa_i00 :::"."::: ) (Set (Var "v"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set ($#k24_aofa_i00 :::"^"::: ) (Set (Var "x"))) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" "A" "," "f") ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"INT-Array":::) "of" "X" "st" (Bool "(" (Bool (Set (Set (Var "a")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_card_1 :::"card"::: ) "X" ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool "(" "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" "holds" (Bool (Set (Set (Var "a")) ($#k1_partfun1 :::"*"::: ) (Set (Var "t"))) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" "A" "," "f") ")" ) ")" )) & (Bool "(" "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" "holds" (Bool (Set ($#k34_valued_1 :::"-"::: ) (Set (Var "t"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") ")" ) & (Bool "(" "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" "holds" (Bool "(" (Bool (Set (Set (Var "t1")) ($#k22_valued_1 :::"(#)"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") & (Bool (Set (Set (Var "t1")) ($#k16_aofa_i00 :::"+"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") & (Bool (Set (Set (Var "t1")) ($#k10_aofa_i00 :::"div"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") & (Bool (Set (Set (Var "t1")) ($#k11_aofa_i00 :::"mod"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") & (Bool (Set ($#k12_aofa_i00 :::"leq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") & (Bool (Set ($#k13_aofa_i00 :::"gt"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f") ")" ) ")" ) ")" ); end; :: deftheorem defines :::"Euclidean"::: AOFA_I00:def 22 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_aofa_i00 :::"Euclidean"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Var "v")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f")))) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k25_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "X")) ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set ($#k23_aofa_i00 :::"."::: ) (Set (Var "v"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k24_aofa_i00 :::"^"::: ) (Set (Var "x"))) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"INT-Array":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Set (Var "a")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_card_1 :::"card"::: ) (Set (Var "X")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool "(" "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "a")) ($#k1_partfun1 :::"*"::: ) (Set (Var "t"))) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) ")" )) & (Bool "(" "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set ($#k34_valued_1 :::"-"::: ) (Set (Var "t"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) & (Bool "(" "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set (Var "t1")) ($#k22_valued_1 :::"(#)"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) & (Bool (Set (Set (Var "t1")) ($#k16_aofa_i00 :::"+"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) & (Bool (Set (Set (Var "t1")) ($#k10_aofa_i00 :::"div"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) & (Bool (Set (Set (Var "t1")) ($#k11_aofa_i00 :::"mod"::: ) (Set (Var "t2"))) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) & (Bool (Set ($#k12_aofa_i00 :::"leq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) & (Bool (Set ($#k13_aofa_i00 :::"gt"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f"))) ")" ) ")" ) ")" ) ")" ))))); definitionlet "A" be ($#l1_unialg_1 :::"preIfWhileAlgebra":::); attr "A" is :::"Euclidean"::: means :: AOFA_I00:def 23 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "ex" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" "A" "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "st" (Bool (Set (Var "f")) "is" ($#v1_aofa_i00 :::"Euclidean"::: ) )))); end; :: deftheorem defines :::"Euclidean"::: AOFA_I00:def 23 : (Bool "for" (Set (Var "A")) "being" ($#l1_unialg_1 :::"preIfWhileAlgebra":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_aofa_i00 :::"Euclidean"::: ) ) "iff" (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "ex" (Set (Var "f")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "st" (Bool (Set (Var "f")) "is" ($#v1_aofa_i00 :::"Euclidean"::: ) )))) ")" )); definitionfunc :::"INT-ElemIns"::: -> ($#v1_finset_1 :::"infinite"::: ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) equals :: AOFA_I00:def 24 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ); end; :: deftheorem defines :::"INT-ElemIns"::: AOFA_I00:def 24 : (Bool (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); definitionmode :::"INT-Exec"::: -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) means :: AOFA_I00:def 25 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))); end; :: deftheorem defines :::"INT-Exec"::: AOFA_I00:def 25 : (Bool "for" (Set (Var "b1")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b1")) "is" ($#m4_aofa_i00 :::"INT-Exec"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))) ")" )); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"INT-ElemIns"::: "X" -> ($#v1_finset_1 :::"infinite"::: ) ($#v1_freealg :::"disjoint_with_NAT"::: ) ($#m1_hidden :::"set"::: ) equals :: AOFA_I00:def 26 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," "X" ")" ")" ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ); end; :: deftheorem defines :::"INT-ElemIns"::: AOFA_I00:def 26 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k27_aofa_i00 :::"INT-ElemIns"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set (Var "X")) ")" ")" ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); mode :::"INT-Exec"::: "of" "x" -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set "(" ($#k27_aofa_i00 :::"INT-ElemIns"::: ) "X" ")" ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" "x" "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) means :: AOFA_I00:def 27 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," "X" ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))); end; :: deftheorem defines :::"INT-Exec"::: AOFA_I00:def 27 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "b3")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set "(" ($#k27_aofa_i00 :::"INT-ElemIns"::: ) (Set (Var "X")) ")" ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "x")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m5_aofa_i00 :::"INT-Exec"::: ) "of" (Set (Var "x"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set (Var "X")) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "v")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))) ")" )))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "c" be ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Const "X")); assume (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Const "c"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_numbers :::"NAT"::: ) )) ; mode :::"INT-Exec"::: "of" "c" "," "T" -> ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," "T" means :: AOFA_I00:def 28 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," "X" ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" "c" ($#k1_partfun1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k17_aofa_i00 :::"**"::: ) "(" "c" "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) "," (Set "(" (Set (Var "e")) ($#k17_aofa_i00 :::"**"::: ) "(" "c" "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))); end; :: deftheorem defines :::"INT-Exec"::: AOFA_I00:def 28 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m6_aofa_i00 :::"INT-Exec"::: ) "of" (Set (Var "c")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set (Var "X")) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b4")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k1_trees_4 :::"root-tree"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" (Set (Var "c")) ($#k1_partfun1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k17_aofa_i00 :::"**"::: ) "(" (Set (Var "c")) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) "," (Set "(" (Set (Var "e")) ($#k17_aofa_i00 :::"**"::: ) "(" (Set (Var "c")) "," (Set ($#k5_numbers :::"NAT"::: ) ) ")" ")" ) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "e")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s")) ")" ) ")" ))))) ")" ))))); theorem :: AOFA_I00:16 (Bool "for" (Set (Var "f")) "being" ($#m4_aofa_i00 :::"INT-Exec"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "v")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f")))))) ; theorem :: AOFA_I00:17 (Bool "for" (Set (Var "f")) "being" ($#m4_aofa_i00 :::"INT-Exec"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "v")) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set (Var "f"))))) ; theorem :: AOFA_I00:18 (Bool "for" (Set (Var "f")) "being" ($#m4_aofa_i00 :::"INT-Exec"::: ) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "t")) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set (Var "f"))))) ; registration cluster -> ($#v1_aofa_i00 :::"Euclidean"::: ) for ($#m4_aofa_i00 :::"INT-Exec"::: ) ; end; theorem :: AOFA_I00:19 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m6_aofa_i00 :::"INT-Exec"::: ) "of" (Set (Var "c")) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "v")) "," (Set (Var "t")) ($#r2_aofa_i00 :::"form_assignment_wrt"::: ) (Set (Var "f"))))))))) ; theorem :: AOFA_I00:20 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m6_aofa_i00 :::"INT-Exec"::: ) "of" (Set (Var "c")) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"INT-Variable":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "v")) "is" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set (Var "f")))))))) ; theorem :: AOFA_I00:21 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m6_aofa_i00 :::"INT-Exec"::: ) "of" (Set (Var "c")) "," (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"INT-Expression":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "t")) "is" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) "," (Set (Var "f")))))))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "c" be ($#m1_subset_1 :::"Enumeration":::) "of" (Set (Const "X")); cluster -> ($#v1_aofa_i00 :::"Euclidean"::: ) for ($#m6_aofa_i00 :::"INT-Exec"::: ) "of" "c" "," "T"; end; registration cluster (Set ($#k7_freealg :::"FreeUnivAlgNSG"::: ) "(" (Set ($#k19_aofa_000 :::"ECIW-signature"::: ) ) "," (Set ($#k26_aofa_i00 :::"INT-ElemIns"::: ) ) ")" ) -> ($#v2_aofa_i00 :::"Euclidean"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) bbbadV2_UNIALG_1() bbbadV3_UNIALG_1() bbbadV4_UNIALG_1() ($#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" ($#v10_aofa_000 :::"degenerated"::: ) ) ($#v2_aofa_i00 :::"Euclidean"::: ) for bbbadL1_UNIALG_1(); end; registrationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k2_zfmisc_1 :::":]"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "A") ($#k2_zfmisc_1 :::":]"::: ) )) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() ($#v13_aofa_000 :::"complying_with_empty-instruction"::: ) ($#v14_aofa_000 :::"complying_with_catenation"::: ) ($#v1_aofa_i00 :::"Euclidean"::: ) for ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" "A" "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," "T"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); :: original: :::"-"::: redefine func :::"-"::: "t" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; :: original: :::"+"::: redefine func "t" :::"+"::: "i" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; :: original: :::"-"::: redefine func "t" :::"-"::: "i" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; :: original: :::"*"::: redefine func "t" :::"*"::: "i" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "t1", "t2" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); :: original: :::"-"::: redefine func "t1" :::"-"::: "t2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; :: original: :::"+"::: redefine func "t1" :::"+"::: "t2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; :: original: :::"(#)"::: redefine func "t1" :::"(#)"::: "t2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "t1", "t2" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); :: original: :::"div"::: redefine func "t1" :::"div"::: "t2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" means :: AOFA_I00:def 29 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))); :: original: :::"mod"::: redefine func "t1" :::"mod"::: "t2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" means :: AOFA_I00:def 30 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))); :: original: :::"leq"::: redefine func :::"leq"::: "(" "t1" "," "t2" ")" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" means :: AOFA_I00:def 31 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" ))); :: original: :::"gt"::: redefine func :::"gt"::: "(" "t1" "," "t2" ")" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" means :: AOFA_I00:def 32 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))); end; :: deftheorem defines :::"div"::: AOFA_I00:def 29 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b7")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k35_aofa_i00 :::"div"::: ) (Set (Var "t2")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b7")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))) ")" )))))); :: deftheorem defines :::"mod"::: AOFA_I00:def 30 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b7")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t1")) ($#k36_aofa_i00 :::"mod"::: ) (Set (Var "t2")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b7")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )))) ")" )))))); :: deftheorem defines :::"leq"::: AOFA_I00:def 31 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b7")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k37_aofa_i00 :::"leq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b7")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" ))) ")" )))))); :: deftheorem defines :::"gt"::: AOFA_I00:def 32 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b7")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k38_aofa_i00 :::"gt"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b7")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_xxreal_0 :::"IFGT"::: ) "(" (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) ")" )))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "t1", "t2" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); :: original: :::"eq"::: redefine func :::"eq"::: "(" "t1" "," "t2" ")" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" means :: AOFA_I00:def 33 (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set "(" "t1" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t2" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))); end; :: deftheorem defines :::"eq"::: AOFA_I00:def 33 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "b7")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k39_aofa_i00 :::"eq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "b7")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set "(" (Set (Var "t1")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t2")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) ")" )))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func :::"."::: "v" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 34 (Set ($#k23_aofa_i00 :::"."::: ) "v"); end; :: deftheorem defines :::"."::: AOFA_I00:def 34 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set ($#k40_aofa_i00 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k23_aofa_i00 :::"."::: ) (Set (Var "v"))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); func "x" :::"^"::: "(" "A" "," "f" ")" -> ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 35 (Set ($#k24_aofa_i00 :::"^"::: ) "x"); end; :: deftheorem defines :::"^"::: AOFA_I00:def 35 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "x")) ($#k41_aofa_i00 :::"^"::: ) "(" (Set (Var "A")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k24_aofa_i00 :::"^"::: ) (Set (Var "x"))))))))); notationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); synonym :::"^"::: "x" for "x" :::"^"::: "(" "A" "," "f" ")" ; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); func :::"."::: "x" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 36 (Set ($#k40_aofa_i00 :::"."::: ) (Set "(" ($#k41_aofa_i00 :::"^"::: ) "x" ")" )); end; :: deftheorem defines :::"."::: AOFA_I00:def 36 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set ($#k42_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k40_aofa_i00 :::"."::: ) (Set "(" ($#k41_aofa_i00 :::"^"::: ) (Set (Var "x")) ")" )))))))); theorem :: AOFA_I00:22 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")))))))))) ; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func :::"."::: "(" "i" "," "A" "," "f" ")" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 37 (Set ($#k25_aofa_i00 :::"."::: ) "(" "i" "," "X" ")" ); end; :: deftheorem defines :::"."::: AOFA_I00:def 37 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k25_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "X")) ")" ))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "v" :::":="::: "t" -> ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 38 (Set ($#k10_subset_1 :::"choose"::: ) "{" (Set (Var "I")) where I "is" ($#m1_subset_1 :::"Element":::) "of" "A" : (Bool "(" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) "A")) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set "f" ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" "v" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" "t" ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ")" ) ")" ) "}" ); end; :: deftheorem defines :::":="::: AOFA_I00:def 38 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set ($#k10_subset_1 :::"choose"::: ) "{" (Set (Var "I")) where I "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k15_funct_7 :::"+*"::: ) "(" (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ")" ) ")" ) "}" )))))))); theorem :: AOFA_I00:23 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set ($#k18_aofa_000 :::"ElementaryInstructions"::: ) (Set (Var "A")))))))))) ; registrationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); cluster (Set "v" ($#k44_aofa_i00 :::":="::: ) "t") -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "v" :::"+="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 39 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) "v" ")" ) ($#k33_aofa_i00 :::"+"::: ) "t" ")" )); func "v" :::"*="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 40 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) "v" ")" ) ($#k34_aofa_i00 :::"(#)"::: ) "t" ")" )); end; :: deftheorem defines :::"+="::: AOFA_I00:def 39 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "v")) ($#k45_aofa_i00 :::"+="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v")) ")" ) ($#k33_aofa_i00 :::"+"::: ) (Set (Var "t")) ")" ))))))))); :: deftheorem defines :::"*="::: AOFA_I00:def 40 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "v")) ($#k46_aofa_i00 :::"*="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v")) ")" ) ($#k34_aofa_i00 :::"(#)"::: ) (Set (Var "t")) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "x" :::":="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 41 (Set (Set "(" "x" ($#k41_aofa_i00 :::"^"::: ) "(" "A" "," "f" ")" ")" ) ($#k44_aofa_i00 :::":="::: ) "t"); end; :: deftheorem defines :::":="::: AOFA_I00:def 41 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k41_aofa_i00 :::"^"::: ) "(" (Set (Var "A")) "," (Set (Var "f")) ")" ")" ) ($#k44_aofa_i00 :::":="::: ) (Set (Var "t")))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "y" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); func "x" :::":="::: "y" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 42 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "y" ")" )); end; :: deftheorem defines :::":="::: AOFA_I00:def 42 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "x" :::":="::: "v" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 43 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) "v" ")" )); end; :: deftheorem defines :::":="::: AOFA_I00:def 43 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k49_aofa_i00 :::":="::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v")) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v", "w" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "v" :::":="::: "w" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 44 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) "w" ")" )); end; :: deftheorem defines :::":="::: AOFA_I00:def 44 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "v")) ($#k50_aofa_i00 :::":="::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "w")) ")" )))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func "x" :::":="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 45 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "f" ")" ")" )); end; :: deftheorem defines :::":="::: AOFA_I00:def 45 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k51_aofa_i00 :::":="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v1", "v2" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); func :::"swap"::: "(" "v1" "," "x" "," "v2" ")" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 46 (Set (Set "(" (Set "(" "x" ($#k49_aofa_i00 :::":="::: ) "v1" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" "v1" ($#k50_aofa_i00 :::":="::: ) "v2" ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" "v2" ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ")" )); end; :: deftheorem defines :::"swap"::: AOFA_I00:def 46 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set ($#k52_aofa_i00 :::"swap"::: ) "(" (Set (Var "v1")) "," (Set (Var "x")) "," (Set (Var "v2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k49_aofa_i00 :::":="::: ) (Set (Var "v1")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "v1")) ($#k50_aofa_i00 :::":="::: ) (Set (Var "v2")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "v2")) ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); let "t" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); func "x" :::"+="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 47 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k33_aofa_i00 :::"+"::: ) "t" ")" )); func "x" :::"*="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 48 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k34_aofa_i00 :::"(#)"::: ) "t" ")" )); func "x" :::"%="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 49 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k36_aofa_i00 :::"mod"::: ) "t" ")" )); func "x" :::"/="::: "t" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 50 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k35_aofa_i00 :::"div"::: ) "t" ")" )); end; :: deftheorem defines :::"+="::: AOFA_I00:def 47 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k53_aofa_i00 :::"+="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k33_aofa_i00 :::"+"::: ) (Set (Var "t")) ")" ))))))))); :: deftheorem defines :::"*="::: AOFA_I00:def 48 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k54_aofa_i00 :::"*="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k34_aofa_i00 :::"(#)"::: ) (Set (Var "t")) ")" ))))))))); :: deftheorem defines :::"%="::: AOFA_I00:def 49 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k55_aofa_i00 :::"%="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set (Var "t")) ")" ))))))))); :: deftheorem defines :::"/="::: AOFA_I00:def 50 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool (Set (Set (Var "x")) ($#k56_aofa_i00 :::"/="::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set (Var "t")) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func "x" :::"+="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 51 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k29_aofa_i00 :::"+"::: ) "i" ")" )); func "x" :::"*="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 52 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k31_aofa_i00 :::"*"::: ) "i" ")" )); func "x" :::"%="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 53 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "f" ")" ")" ) ")" )); func "x" :::"/="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 54 (Set "x" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "f" ")" ")" ) ")" )); func "x" :::"div"::: "i" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 55 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "f" ")" ")" )); end; :: deftheorem defines :::"+="::: AOFA_I00:def 51 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k57_aofa_i00 :::"+="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k29_aofa_i00 :::"+"::: ) (Set (Var "i")) ")" ))))))))); :: deftheorem defines :::"*="::: AOFA_I00:def 52 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k58_aofa_i00 :::"*="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k31_aofa_i00 :::"*"::: ) (Set (Var "i")) ")" ))))))))); :: deftheorem defines :::"%="::: AOFA_I00:def 53 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k59_aofa_i00 :::"%="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ")" ) ")" ))))))))); :: deftheorem defines :::"/="::: AOFA_I00:def 54 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k60_aofa_i00 :::"/="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ")" ) ")" ))))))))); :: deftheorem defines :::"div"::: AOFA_I00:def 55 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k61_aofa_i00 :::"div"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func "v" :::":="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 56 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "f" ")" ")" )); end; :: deftheorem defines :::":="::: AOFA_I00:def 56 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "v")) ($#k62_aofa_i00 :::":="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "f")) ")" ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "v" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "f")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func "v" :::"+="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 57 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) "v" ")" ) ($#k29_aofa_i00 :::"+"::: ) "i" ")" )); func "v" :::"*="::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 58 (Set "v" ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) "v" ")" ) ($#k31_aofa_i00 :::"*"::: ) "i" ")" )); end; :: deftheorem defines :::"+="::: AOFA_I00:def 57 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "v")) ($#k63_aofa_i00 :::"+="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v")) ")" ) ($#k29_aofa_i00 :::"+"::: ) (Set (Var "i")) ")" ))))))))); :: deftheorem defines :::"*="::: AOFA_I00:def 58 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "v")) ($#k64_aofa_i00 :::"*="::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v")) ")" ) ($#k31_aofa_i00 :::"*"::: ) (Set (Var "i")) ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "t1" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "g")); func "t1" :::"is_odd"::: -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 59 (Set "b" ($#k47_aofa_i00 :::":="::: ) (Set "(" "t1" ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Num 2) "," "A" "," "g" ")" ")" ) ")" )); func "t1" :::"is_even"::: -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 60 (Set "b" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" "t1" ($#k29_aofa_i00 :::"+"::: ) (Num 1) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Num 2) "," "A" "," "g" ")" ")" ) ")" )); let "t2" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "g")); func "t1" :::"leq"::: "t2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 61 (Set "b" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k37_aofa_i00 :::"leq"::: ) "(" "t1" "," "t2" ")" ")" )); func "t1" :::"gt"::: "t2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 62 (Set "b" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k38_aofa_i00 :::"gt"::: ) "(" "t1" "," "t2" ")" ")" )); func "t1" :::"eq"::: "t2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 63 (Set "b" ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k39_aofa_i00 :::"eq"::: ) "(" "t1" "," "t2" ")" ")" )); end; :: deftheorem defines :::"is_odd"::: AOFA_I00:def 59 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t1")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "t1")) ($#k65_aofa_i00 :::"is_odd"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set (Var "t1")) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Num 2) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ) ")" )))))))); :: deftheorem defines :::"is_even"::: AOFA_I00:def 60 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t1")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "t1")) ($#k66_aofa_i00 :::"is_even"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" (Set (Var "t1")) ($#k29_aofa_i00 :::"+"::: ) (Num 1) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Num 2) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ) ")" )))))))); :: deftheorem defines :::"leq"::: AOFA_I00:def 61 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "t1")) ($#k67_aofa_i00 :::"leq"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k37_aofa_i00 :::"leq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" )))))))); :: deftheorem defines :::"gt"::: AOFA_I00:def 62 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "t1")) ($#k68_aofa_i00 :::"gt"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k38_aofa_i00 :::"gt"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" )))))))); :: deftheorem defines :::"eq"::: AOFA_I00:def 63 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "t1")) ($#k69_aofa_i00 :::"eq"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k47_aofa_i00 :::":="::: ) (Set "(" ($#k39_aofa_i00 :::"eq"::: ) "(" (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" )))))))); notationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "t1", "t2" be ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Const "A")) "," (Set (Const "g")); synonym "t2" :::"geq"::: "t1" for "t1" :::"leq"::: "t2"; synonym "t2" :::"lt"::: "t1" for "t1" :::"gt"::: "t2"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "v1", "v2" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "g")); func "v1" :::"leq"::: "v2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 64 (Set (Set "(" ($#k40_aofa_i00 :::"."::: ) "v1" ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) "v2" ")" )); func "v1" :::"gt"::: "v2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 65 (Set (Set "(" ($#k40_aofa_i00 :::"."::: ) "v1" ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) "v2" ")" )); end; :: deftheorem defines :::"leq"::: AOFA_I00:def 64 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "v1")) ($#k70_aofa_i00 :::"leq"::: ) (Set (Var "v2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v1")) ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v2")) ")" )))))))); :: deftheorem defines :::"gt"::: AOFA_I00:def 65 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool (Set (Set (Var "v1")) ($#k71_aofa_i00 :::"gt"::: ) (Set (Var "v2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v1")) ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k40_aofa_i00 :::"."::: ) (Set (Var "v2")) ")" )))))))); notationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "v1", "v2" be ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Const "A")) "," (Set (Const "g")); synonym "v2" :::"geq"::: "v1" for "v1" :::"leq"::: "v2"; synonym "v2" :::"lt"::: "v1" for "v1" :::"gt"::: "v2"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "x1" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "g")); func "x1" :::"is_odd"::: -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 66 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k65_aofa_i00 :::"is_odd"::: ) ); func "x1" :::"is_even"::: -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 67 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k66_aofa_i00 :::"is_even"::: ) ); let "x2" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "g")); func "x1" :::"leq"::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 68 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"gt"::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 69 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); end; :: deftheorem defines :::"is_odd"::: AOFA_I00:def 66 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x1")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool (Set (Set (Var "x1")) ($#k72_aofa_i00 :::"is_odd"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k65_aofa_i00 :::"is_odd"::: ) ))))))); :: deftheorem defines :::"is_even"::: AOFA_I00:def 67 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x1")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool (Set (Set (Var "x1")) ($#k73_aofa_i00 :::"is_even"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k66_aofa_i00 :::"is_even"::: ) ))))))); :: deftheorem defines :::"leq"::: AOFA_I00:def 68 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool (Set (Set (Var "x1")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"gt"::: AOFA_I00:def 69 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool (Set (Set (Var "x1")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); notationlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "x1", "x2" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "g")); synonym "x2" :::"geq"::: "x1" for "x1" :::"leq"::: "x2"; synonym "x2" :::"lt"::: "x1" for "x1" :::"gt"::: "x2"; end; definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "g" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Const "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); let "x" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "g")); let "i" be ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) ; func "x" :::"leq"::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 70 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "g" ")" ")" )); func "x" :::"geq"::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 71 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k67_aofa_i00 :::"geq"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "g" ")" ")" )); func "x" :::"gt"::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 72 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "g" ")" ")" )); func "x" :::"lt"::: "i" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 73 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k68_aofa_i00 :::"lt"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "g" ")" ")" )); func "x" :::"/"::: "i" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "g" equals :: AOFA_I00:def 74 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x" ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" "i" "," "A" "," "g" ")" ")" )); end; :: deftheorem defines :::"leq"::: AOFA_I00:def 70 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ))))))))); :: deftheorem defines :::"geq"::: AOFA_I00:def 71 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k67_aofa_i00 :::"geq"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ))))))))); :: deftheorem defines :::"gt"::: AOFA_I00:def 72 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k68_aofa_i00 :::"gt"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ))))))))); :: deftheorem defines :::"lt"::: AOFA_I00:def 73 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k68_aofa_i00 :::"lt"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ))))))))); :: deftheorem defines :::"/"::: AOFA_I00:def 74 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k80_aofa_i00 :::"/"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k43_aofa_i00 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "A")) "," (Set (Var "g")) ")" ")" ))))))))); definitionlet "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "x1", "x2" be ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Const "f")); func "x1" :::"+="::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 75 (Set "x1" ($#k53_aofa_i00 :::"+="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"*="::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 76 (Set "x1" ($#k54_aofa_i00 :::"*="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"%="::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 77 (Set "x1" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" ) ")" )); func "x1" :::"/="::: "x2" -> ($#v15_aofa_000 :::"absolutely-terminating"::: ) ($#m1_subset_1 :::"Element":::) "of" "A" equals :: AOFA_I00:def 78 (Set "x1" ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" ) ")" )); func "x1" :::"+"::: "x2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 79 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k33_aofa_i00 :::"+"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"*"::: "x2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 80 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k34_aofa_i00 :::"(#)"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"mod"::: "x2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 81 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); func "x1" :::"div"::: "x2" -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" "A" "," "f" equals :: AOFA_I00:def 82 (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) "x1" ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) "x2" ")" )); end; :: deftheorem defines :::"+="::: AOFA_I00:def 75 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k81_aofa_i00 :::"+="::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k53_aofa_i00 :::"+="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"*="::: AOFA_I00:def 76 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k54_aofa_i00 :::"*="::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"%="::: AOFA_I00:def 77 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k83_aofa_i00 :::"%="::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" ) ")" )))))))); :: deftheorem defines :::"/="::: AOFA_I00:def 78 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k84_aofa_i00 :::"/="::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" ) ")" )))))))); :: deftheorem defines :::"+"::: AOFA_I00:def 79 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k85_aofa_i00 :::"+"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k33_aofa_i00 :::"+"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"*"::: AOFA_I00:def 80 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k86_aofa_i00 :::"*"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k34_aofa_i00 :::"(#)"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"mod"::: AOFA_I00:def 81 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k87_aofa_i00 :::"mod"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k36_aofa_i00 :::"mod"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); :: deftheorem defines :::"div"::: AOFA_I00:def 82 : (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool (Set (Set (Var "x1")) ($#k88_aofa_i00 :::"div"::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x1")) ")" ) ($#k35_aofa_i00 :::"div"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x2")) ")" )))))))); theorem :: AOFA_I00:24 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "v")) "being" ($#m2_aofa_i00 :::"INT-Variable"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "v")) ($#k44_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:25 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k51_aofa_i00 :::":="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "i"))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k51_aofa_i00 :::":="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:26 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:27 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:28 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k57_aofa_i00 :::"+="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k57_aofa_i00 :::"+="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:29 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k53_aofa_i00 :::"+="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k53_aofa_i00 :::"+="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:30 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k81_aofa_i00 :::"+="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k81_aofa_i00 :::"+="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:31 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k58_aofa_i00 :::"*="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k58_aofa_i00 :::"*="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:32 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k54_aofa_i00 :::"*="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k54_aofa_i00 :::"*="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:33 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:34 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ) ")" )))))))) ; theorem :: AOFA_I00:35 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:36 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k76_aofa_i00 :::"leq"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k77_aofa_i00 :::"geq"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i"))) ")" ")" )))))))) ; theorem :: AOFA_I00:37 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"geq"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k74_aofa_i00 :::"geq"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) ")" ")" ))))))) ; theorem :: AOFA_I00:38 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ) ")" )))))))) ; theorem :: AOFA_I00:39 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"lt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"lt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"lt"::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ) ")" ))))))) ; theorem :: AOFA_I00:40 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k78_aofa_i00 :::"gt"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "i"))) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k79_aofa_i00 :::"lt"::: ) (Set (Var "i")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) ")" ")" )))))))) ; theorem :: AOFA_I00:41 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"gt"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))))) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"lt"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k75_aofa_i00 :::"lt"::: ) (Set (Var "y")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) ")" ")" ))))))) ; theorem :: AOFA_I00:42 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k59_aofa_i00 :::"%="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k59_aofa_i00 :::"%="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:43 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k55_aofa_i00 :::"%="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k55_aofa_i00 :::"%="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:44 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k83_aofa_i00 :::"%="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k83_aofa_i00 :::"%="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:45 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k60_aofa_i00 :::"/="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_int_1 :::"div"::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k60_aofa_i00 :::"/="::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:46 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k56_aofa_i00 :::"/="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k56_aofa_i00 :::"/="::: ) (Set (Var "t")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" )))))))) ; theorem :: AOFA_I00:47 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "f")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k84_aofa_i00 :::"/="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_int_1 :::"div"::: ) (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k84_aofa_i00 :::"/="::: ) (Set (Var "y")) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:48 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k65_aofa_i00 :::"is_odd"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k6_int_1 :::"mod"::: ) (Num 2))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k66_aofa_i00 :::"is_even"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ($#k6_int_1 :::"mod"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k65_aofa_i00 :::"is_odd"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k66_aofa_i00 :::"is_even"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ) ")" ))))))) ; theorem :: AOFA_I00:49 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k6_int_1 :::"mod"::: ) (Num 2))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k73_aofa_i00 :::"is_even"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ($#k6_int_1 :::"mod"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "z")))) ")" ) ")" ))))))) ; theorem :: AOFA_I00:50 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) "is" ($#v1_abian :::"odd"::: ) )) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k65_aofa_i00 :::"is_odd"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k65_aofa_i00 :::"is_odd"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) "is" ($#v1_abian :::"odd"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) "is" ($#v1_abian :::"even"::: ) )) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k66_aofa_i00 :::"is_even"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "t")) ($#k66_aofa_i00 :::"is_even"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "t")) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) "is" ($#v1_abian :::"even"::: ) ) ")" ")" ))))))) ; theorem :: AOFA_I00:51 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) "is" ($#v1_abian :::"odd"::: ) )) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) "is" ($#v1_abian :::"odd"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) "is" ($#v1_abian :::"even"::: ) )) "implies" (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k73_aofa_i00 :::"is_even"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) ")" & "(" (Bool (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "x")) ($#k73_aofa_i00 :::"is_even"::: ) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "implies" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) "is" ($#v1_abian :::"even"::: ) ) ")" ")" ))))))) ; scheme :: AOFA_I00:sch 1 ForToIteration{ F1() -> ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ), F4() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F5() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F6() -> ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set F3 "(" ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ), F7() -> ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set F6 "(" ")" ), F8() -> ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set F6 "(" ")" ), F9() -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ), F10() -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set F1 "(" ")" ) "," (Set F6 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "(" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" )]) & "(" (Bool (Bool (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )))) "implies" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1))) ")" & "(" (Bool (Bool (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )))) "implies" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" ))) ")" & (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ))) ")" ) provided (Bool (Set F5 "(" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set F7 "(" ")" ) ($#k47_aofa_i00 :::":="::: ) (Set F10 "(" ")" ) ")" ) "," (Set "(" (Set F7 "(" ")" ) ($#k74_aofa_i00 :::"leq"::: ) (Set F8 "(" ")" ) ")" ) "," (Set "(" (Set F7 "(" ")" ) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set F4 "(" ")" ) ")" )) and (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set "(" (Set F7 "(" ")" ) ($#k47_aofa_i00 :::":="::: ) (Set F10 "(" ")" ) ")" ) ")" )]) and (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set F4 "(" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set F7 "(" ")" ) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) ")" ) ")" )]) & (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set F7 "(" ")" ) ($#k74_aofa_i00 :::"leq"::: ) (Set F8 "(" ")" ) ")" ) ")" )]) ")" )) and (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F4 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" ))) & (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F4 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ))) ")" )) and (Bool "(" (Bool (Set F8 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F7 "(" ")" )) & (Bool (Set F8 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F3 "(" ")" )) & (Bool (Set F7 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F3 "(" ")" )) ")" ) proof end; scheme :: AOFA_I00:sch 2 ForDowntoIteration{ F1() -> ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ), F4() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F5() -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ), F6() -> ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set F1 "(" ")" ) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set F3 "(" ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ), F7() -> ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set F6 "(" ")" ), F8() -> ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set F6 "(" ")" ), F9() -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ), F10() -> ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set F1 "(" ")" ) "," (Set F6 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "(" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" )]) & "(" (Bool (Bool (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )))) "implies" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" & "(" (Bool (Bool (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )))) "implies" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set F10 "(" ")" ) ($#k3_funct_2 :::"."::: ) (Set F9 "(" ")" ))) ")" & (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set F5 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set F9 "(" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ))) ")" ) provided (Bool (Set F5 "(" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set F7 "(" ")" ) ($#k47_aofa_i00 :::":="::: ) (Set F10 "(" ")" ) ")" ) "," (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set F8 "(" ")" ) ")" ) ($#k67_aofa_i00 :::"leq"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set F7 "(" ")" ) ")" ) ")" ) "," (Set "(" (Set F7 "(" ")" ) ($#k57_aofa_i00 :::"+="::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) "," (Set F4 "(" ")" ) ")" )) and (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set F9 "(" ")" ) "," (Set "(" (Set F7 "(" ")" ) ($#k47_aofa_i00 :::":="::: ) (Set F10 "(" ")" ) ")" ) ")" )]) and (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set F4 "(" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set F7 "(" ")" ) ($#k57_aofa_i00 :::"+="::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ")" )]) & (Bool P1[(Set (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set F8 "(" ")" ) ($#k74_aofa_i00 :::"leq"::: ) (Set F7 "(" ")" ) ")" ) ")" )]) ")" )) and (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set F2 "(" ")" ) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool P1[(Set (Var "s"))])) "holds" (Bool "(" (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F4 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set F7 "(" ")" ))) & (Bool (Set (Set "(" (Set F6 "(" ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set F4 "(" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set F8 "(" ")" ))) ")" )) and (Bool "(" (Bool (Set F8 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F7 "(" ")" )) & (Bool (Set F8 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F3 "(" ")" )) & (Bool (Set F7 "(" ")" ) ($#r1_hidden :::"<>"::: ) (Set F3 "(" ")" )) ")" ) proof end; begin theorem :: AOFA_I00:52 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" )) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i")))) ")" ) ")" )) "holds" (Bool (Set (Var "g")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" )) "," (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) ")" ))))))))) ; theorem :: AOFA_I00:53 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" )) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" ) ")" ) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Var "g")) ($#r4_aofa_000 :::"iteration_terminates_for"::: ) (Set (Set "(" (Set (Var "I")) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" )) "," (Set (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) ")" )))))))))) ; theorem :: AOFA_I00:54 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (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 "g")))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" )) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i")))) ")" ) ")" )) "holds" (Bool (Set ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set (Var "I")) ")" ) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")))))))))))) ; theorem :: AOFA_I00:55 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#m3_aofa_i00 :::"INT-Expression"::: ) "of" (Set (Var "A")) "," (Set (Var "f")) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "I")) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")) "," (Set (Var "P")))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" )) & (Bool "(" "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "s")) "," (Set (Var "I")) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "i")))) ")" ) ")" ) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Set (Var "i")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t"))) "," (Set (Var "g"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Var "I")) "," (Set (Var "g"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n"))) "," (Set (Var "g"))) & (Bool (Set (Var "P")) ($#r7_aofa_000 :::"is_invariant_wrt"::: ) (Set (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1)) "," (Set (Var "g")))) "holds" (Bool (Set ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k47_aofa_i00 :::":="::: ) (Set (Var "t")) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set (Var "I")) ")" ) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")) "," (Set (Var "P"))))))))))))) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::); let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ); let "f" be ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Const "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Const "T")); let "s" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Const "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ); let "I" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "A")); :: original: :::"."::: redefine func "f" :::"."::: "(" "s" "," "I" ")" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ); end; theorem :: AOFA_I00:56 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "s")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 4)) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "s")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "s")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "i")) ")" ) ")" ")" )) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")))))))) ; theorem :: AOFA_I00:57 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "s")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 4)) ")" ))) "holds" (Bool "for" (Set (Var "q")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "N")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "q")) "," (Set "(" (Set "(" (Set (Var "s")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "s")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "N")) ($#k9_newton :::"!"::: ) ))))))))) ; theorem :: AOFA_I00:58 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "s")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 4)) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "s")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "s")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ")" )) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")))))))) ; theorem :: AOFA_I00:59 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "s")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Num 4)) ")" ))) "holds" (Bool "for" (Set (Var "q")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "N")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "q")) "," (Set "(" (Set "(" (Set (Var "s")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "s")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ")" ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "N"))))))))))) ; theorem :: AOFA_I00:60 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 4)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 5)) ")" ))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k51_aofa_i00 :::":="::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "x")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k81_aofa_i00 :::"+="::: ) (Set (Var "z")) ")" ) ")" ) ")" ")" )) ($#r5_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")))))))) ; theorem :: AOFA_I00:61 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "i")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 4)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 5)) ")" ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set "(" (Set "(" (Set (Var "x")) ($#k51_aofa_i00 :::":="::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k17_aofa_000 :::"for-do"::: ) "(" (Set "(" (Set (Var "i")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "i")) ($#k74_aofa_i00 :::"leq"::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "i")) ($#k57_aofa_i00 :::"+="::: ) (Num 1) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "x")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k81_aofa_i00 :::"+="::: ) (Set (Var "z")) ")" ) ")" ) ")" ")" ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "N"))))))))))) ; theorem :: AOFA_I00:62 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "y")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "x")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "z")) ($#k83_aofa_i00 :::"%="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "z")) ")" ) ")" ) ")" ) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")) "," "{" (Set (Var "s")) where s "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) : (Bool "(" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )))))) ; theorem :: AOFA_I00:63 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "y")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "x")) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "z")) ($#k83_aofa_i00 :::"%="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "z")) ")" ) ")" ) ")" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k6_nat_d :::"gcd"::: ) (Set (Var "m"))))))))))) ; theorem :: AOFA_I00:64 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool (Set ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "y")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k32_aofa_i00 :::"-"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k15_aofa_000 :::"if-then"::: ) "(" (Set "(" (Set (Var "z")) ($#k79_aofa_i00 :::"lt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set (Var "z")) ($#k58_aofa_i00 :::"*="::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "z")) ")" ) ")" ) ")" ) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")) "," "{" (Set (Var "s")) where s "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) : (Bool "(" (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )))))) ; theorem :: AOFA_I00:65 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "y")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "z")) ($#k47_aofa_i00 :::":="::: ) (Set "(" (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k32_aofa_i00 :::"-"::: ) (Set "(" ($#k42_aofa_i00 :::"."::: ) (Set (Var "y")) ")" ) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k15_aofa_000 :::"if-then"::: ) "(" (Set "(" (Set (Var "z")) ($#k79_aofa_i00 :::"lt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set (Var "z")) ($#k58_aofa_i00 :::"*="::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "y")) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "y")) ($#k48_aofa_i00 :::":="::: ) (Set (Var "z")) ")" ) ")" ) ")" ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k6_nat_d :::"gcd"::: ) (Set (Var "m"))))))))))) ; theorem :: AOFA_I00:66 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "m")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "y")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "m")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k15_aofa_000 :::"if-then"::: ) "(" (Set "(" (Set (Var "m")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) "," (Set "(" (Set (Var "y")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "m")) ($#k60_aofa_i00 :::"/="::: ) (Num 2) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ) ")" ")" )) ($#r6_aofa_000 :::"is_terminating_wrt"::: ) (Set (Var "g")) "," "{" (Set (Var "s")) where s "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) : (Bool (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) "}" )))))) ; theorem :: AOFA_I00:67 (Bool "for" (Set (Var "A")) "being" ($#v2_aofa_i00 :::"Euclidean"::: ) ($#l1_unialg_1 :::"preIfWhileAlgebra":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#v1_aofa_i00 :::"Euclidean"::: ) ($#m1_aofa_000 :::"ExecutionFunction"::: ) "of" (Set (Var "A")) "," (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) "," (Set (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ")" ) ($#k1_aofa_i00 :::"\"::: ) "(" (Set (Var "b")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "m")) "being" ($#m1_aofa_i00 :::"Variable"::: ) "of" (Set (Var "g")) "st" (Bool (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Num 3)) ")" ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k4_numbers :::"INT"::: ) ) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k89_aofa_i00 :::"."::: ) "(" (Set (Var "s")) "," (Set "(" (Set "(" (Set (Var "y")) ($#k51_aofa_i00 :::":="::: ) (Num 1) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" ($#k16_aofa_000 :::"while"::: ) "(" (Set "(" (Set (Var "m")) ($#k78_aofa_i00 :::"gt"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k15_aofa_000 :::"if-then"::: ) "(" (Set "(" (Set (Var "m")) ($#k72_aofa_i00 :::"is_odd"::: ) ")" ) "," (Set "(" (Set (Var "y")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "m")) ($#k60_aofa_i00 :::"/="::: ) (Num 2) ")" ) ")" ) ($#k13_aofa_000 :::"\;"::: ) (Set "(" (Set (Var "x")) ($#k82_aofa_i00 :::"*="::: ) (Set (Var "x")) ")" ) ")" ) ")" ")" ) ")" ) ")" ")" ) ($#k21_aofa_i00 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k21_aofa_i00 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n"))))))))))) ;