:: AMISTD_1 semantic presentation begin registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))); let "s" be ($#m1_hidden :::"State":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set "the" ($#u1_extpro_1 :::"Execution"::: ) "of" "S") ($#k3_funct_2 :::"."::: ) "i" ")" ) ($#k1_funct_1 :::"."::: ) "s") -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) for ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" "N"; end; definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "T" be ($#m1_subset_1 :::"InsType":::) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))); attr "T" is :::"jump-only"::: means :: AMISTD_1:def 1 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" "S" (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" "S" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Instruction":::) "of" "S" "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) "T") & (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set ($#k8_struct_0 :::"Data-Locations"::: ) ))) "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "o"))))))); end; :: deftheorem defines :::"jump-only"::: AMISTD_1:def 1 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"InsType":::) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_amistd_1 :::"jump-only"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"Object":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set ($#k8_struct_0 :::"Data-Locations"::: ) ))) "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "I")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "o"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "o"))))))) ")" )))); definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "I" be ($#m1_subset_1 :::"Instruction":::) "of" (Set (Const "S")); attr "I" is :::"jump-only"::: means :: AMISTD_1:def 2 (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) "I") "is" ($#v1_amistd_1 :::"jump-only"::: ) ); end; :: deftheorem defines :::"jump-only"::: AMISTD_1:def 2 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v2_amistd_1 :::"jump-only"::: ) ) "iff" (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "I"))) "is" ($#v1_amistd_1 :::"jump-only"::: ) ) ")" )))); definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "l" be ($#m1_hidden :::"Nat":::); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))); func :::"NIC"::: "(" "i" "," "l" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMISTD_1:def 3 "{" (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" "i" "," (Set (Var "ss")) ")" ")" ) ")" ) where ss "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) "S" ")" )) : (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "ss"))) ($#r1_hidden :::"="::: ) "l") "}" ; end; :: deftheorem defines :::"NIC"::: AMISTD_1:def 3 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "S"))) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "ss")) ")" ")" ) ")" ) where ss "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) (Set (Var "S")) ")" )) : (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "ss"))) ($#r1_hidden :::"="::: ) (Set (Var "l"))) "}" ))))); registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))); let "l" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k1_amistd_1 :::"NIC"::: ) "(" "i" "," "l" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))); func :::"JUMP"::: "i" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMISTD_1:def 4 (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set "(" ($#k1_amistd_1 :::"NIC"::: ) "(" "i" "," (Set (Var "l")) ")" ")" ) where l "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ); end; :: deftheorem defines :::"JUMP"::: AMISTD_1:def 4 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "S"))) "holds" (Bool (Set ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set "(" ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" ")" ) where l "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ))))); definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "l" be ($#m1_hidden :::"Nat":::); func :::"SUCC"::: "(" "l" "," "S" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMISTD_1:def 5 (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" (Set "(" ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," "l" ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i")) ")" ) ")" ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") : (Bool verum) "}" ); end; :: deftheorem defines :::"SUCC"::: AMISTD_1:def 5 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "l")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" (Set "(" ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i")) ")" ) ")" ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "S"))) : (Bool verum) "}" ))))); theorem :: AMISTD_1:1 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool "(" "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "l")) ($#k6_domain_1 :::"}"::: ) )) ")" )) "holds" (Bool (Set ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i"))) "is" ($#v1_xboole_0 :::"empty"::: ) )))) ; theorem :: AMISTD_1:2 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "il")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "i")) "is" ($#v2_extpro_1 :::"halting"::: ) )) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "il")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "il")) ($#k6_domain_1 :::"}"::: ) )))))) ; begin definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); attr "S" is :::"standard"::: means :: AMISTD_1:def 6 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) "," "S" ")" )) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"standard"::: AMISTD_1:def 6 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_amistd_1 :::"standard"::: ) ) "iff" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) "," (Set (Var "S")) ")" )) ")" ) ")" )) ")" )) ")" ))); theorem :: AMISTD_1:3 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_amistd_1 :::"standard"::: ) ) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" )) & (Bool "(" "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" ))) "holds" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) ")" ) ")" )) ")" ))) ; begin definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; func :::"STC"::: "N" -> ($#v1_extpro_1 :::"strict"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" "N" means :: AMISTD_1:def 7 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set ($#k4_domain_1 :::"["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) "," (Set ($#k4_domain_1 :::"["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set "the" ($#u1_memstr_0 :::"Object-Kind"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool (Set "the" ($#u2_memstr_0 :::"ValuesF"::: ) "of" it) ($#r8_pboole :::"="::: ) (Set "N" ($#k8_funcop_1 :::"-->"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) it ")" ) ")" ) "," (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) it ")" ) ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) it ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set "(" (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set "the" ($#u1_extpro_1 :::"Execution"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k4_domain_1 :::"["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "f")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k4_domain_1 :::"["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) it ")" ) ")" ) ")" ) ")" ))) ")" )) ")" ); end; :: deftheorem defines :::"STC"::: AMISTD_1:def 7 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#v1_extpro_1 :::"strict"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set ($#k4_domain_1 :::"["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) "," (Set ($#k4_domain_1 :::"["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set "the" ($#u1_memstr_0 :::"Object-Kind"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool (Set "the" ($#u2_memstr_0 :::"ValuesF"::: ) "of" (Set (Var "b2"))) ($#r8_pboole :::"="::: ) (Set (Set (Var "N")) ($#k8_funcop_1 :::"-->"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) (Set (Var "b2")) ")" ) ")" ) "," (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) (Set (Var "b2")) ")" ) ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) (Set (Var "b2")) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set "(" (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set "the" ($#u1_extpro_1 :::"Execution"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k4_domain_1 :::"["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "f")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k4_domain_1 :::"["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k4_domain_1 :::"]"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k2_memstr_0 :::"the_Values_of"::: ) (Set (Var "b2")) ")" ) ")" ) ")" ) ")" ))) ")" )) ")" ) ")" ))); registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_struct_0 :::"finite"::: ) ($#v1_extpro_1 :::"strict"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v1_extpro_1 :::"strict"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v1_extpro_1 :::"strict"::: ) ; end; theorem :: AMISTD_1:4 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "i")) "is" ($#v2_extpro_1 :::"halting"::: ) ))) ; theorem :: AMISTD_1:5 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool "not" (Bool (Set (Var "i")) "is" ($#v2_extpro_1 :::"halting"::: ) )))) ; theorem :: AMISTD_1:6 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" )) "holds" (Bool "(" (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: AMISTD_1:7 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "holds" (Bool (Set (Var "i")) "is" ($#v2_amistd_1 :::"jump-only"::: ) ))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v4_compos_0 :::"ins-loc-free"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) "N" ")" )); end; theorem :: AMISTD_1:8 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set (Var "z")))) "holds" (Bool (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "l")) "," (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "z")) "," (Set "(" (Set (Var "z")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k2_tarski :::"}"::: ) ))))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#v1_extpro_1 :::"strict"::: ) ($#v3_amistd_1 :::"standard"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#v1_extpro_1 :::"strict"::: ) ($#v3_extpro_1 :::"halting"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v3_amistd_1 :::"standard"::: ) for ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" "N"; end; theorem :: AMISTD_1:9 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"IC"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" )))))) ; theorem :: AMISTD_1:10 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" )) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set (Var "l")) ")" ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: AMISTD_1:11 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "l")) "," (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "l")) "," (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set (Var "l")) ")" ) ($#k2_tarski :::"}"::: ) )))) ; definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "i" be ($#m1_subset_1 :::"Instruction":::) "of" (Set (Const "S")); attr "i" is :::"sequential"::: means :: AMISTD_1:def 8 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" "S" "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" "i" "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"IC"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" )))); end; :: deftheorem defines :::"sequential"::: AMISTD_1:def 8 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "i")) "is" ($#v4_amistd_1 :::"sequential"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"IC"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" )))) ")" )))); theorem :: AMISTD_1:12 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "il")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "i")) "is" ($#v4_amistd_1 :::"sequential"::: ) )) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "il")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set (Var "il")) ")" ) ($#k1_tarski :::"}"::: ) )))))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v4_amistd_1 :::"sequential"::: ) -> ($#~v2_extpro_1 "non" ($#v2_extpro_1 :::"halting"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S"); cluster ($#v2_extpro_1 :::"halting"::: ) -> ($#~v4_amistd_1 "non" ($#v4_amistd_1 :::"sequential"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S"); end; theorem :: AMISTD_1:13 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "T")) "st" (Bool (Bool (Bool "not" (Set ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "i")) "is" ($#v4_amistd_1 :::"sequential"::: ) ))))) ; begin definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "F" be ($#m1_hidden :::"preProgram":::) "of" (Set (Const "S")); attr "F" is :::"really-closed"::: means :: AMISTD_1:def 9 (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F"))) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set "(" "F" ($#k7_partfun1 :::"/."::: ) (Set (Var "l")) ")" ) "," (Set (Var "l")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F"))); end; :: deftheorem defines :::"really-closed"::: AMISTD_1:def 9 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"preProgram":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v5_amistd_1 :::"really-closed"::: ) ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set "(" (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "l")) ")" ) "," (Set (Var "l")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) ")" )))); definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "F" be (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); attr "F" is :::"paraclosed"::: means :: AMISTD_1:def 10 (Bool "for" (Set (Var "s")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v5_memstr_0 :::"-started"::: ) ($#m1_hidden :::"State":::) "of" "S" (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" "S" "st" (Bool (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set (Var "P")) "," (Set (Var "s")) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F"))))); end; :: deftheorem defines :::"paraclosed"::: AMISTD_1:def 10 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v6_amistd_1 :::"paraclosed"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v5_memstr_0 :::"-started"::: ) ($#m1_hidden :::"State":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set (Var "P")) "," (Set (Var "s")) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))))) ")" )))); definitionlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "F" be (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Const "S"))) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); attr "F" is :::"parahalting"::: means :: AMISTD_1:def 11 (Bool "for" (Set (Var "s")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v5_memstr_0 :::"-started"::: ) ($#m1_hidden :::"State":::) "of" "S" (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" "S" "st" (Bool (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool (Set (Var "P")) ($#r1_extpro_1 :::"halts_on"::: ) (Set (Var "s"))))); end; :: deftheorem defines :::"parahalting"::: AMISTD_1:def 11 : (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v7_amistd_1 :::"parahalting"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v5_memstr_0 :::"-started"::: ) ($#m1_hidden :::"State":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool (Set (Var "P")) ($#r1_extpro_1 :::"halts_on"::: ) (Set (Var "s"))))) ")" )))); theorem :: AMISTD_1:14 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"preProgram":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v5_amistd_1 :::"really-closed"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set (Var "F")) "," (Set (Var "s")) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")))))) ")" )))) ; theorem :: AMISTD_1:15 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "F")) "is" ($#v5_amistd_1 :::"really-closed"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "F")) "is" ($#v6_amistd_1 :::"paraclosed"::: ) )))) ; theorem :: AMISTD_1:16 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k2_compos_1 :::"halt"::: ) (Set (Var "S")) ")" )) "is" ($#v5_amistd_1 :::"really-closed"::: ) ))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_afinsq_1 :::"initial"::: ) ($#v5_amistd_1 :::"really-closed"::: ) -> (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v6_amistd_1 :::"paraclosed"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_afinsq_1 :::"initial"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v5_ordinal1 :::"T-Sequence-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_afinsq_1 :::"initial"::: ) ($#v3_compos_1 :::"halt-ending"::: ) ($#v4_compos_1 :::"unique-halt"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v3_amistd_1 :::"standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v5_ordinal1 :::"T-Sequence-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_afinsq_1 :::"initial"::: ) ($#v3_compos_1 :::"halt-ending"::: ) ($#v4_compos_1 :::"unique-halt"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: AMISTD_1:17 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k1_extpro_1 :::"Trivial-AMI"::: ) (Set (Var "N")) ")" ) "holds" (Bool (Set (Var "i")) "is" ($#v2_extpro_1 :::"halting"::: ) ))) ; theorem :: AMISTD_1:18 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set "(" ($#k1_extpro_1 :::"Trivial-AMI"::: ) (Set (Var "N")) ")" )) "holds" (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; begin theorem :: AMISTD_1:19 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k2_amistd_1 :::"JUMP"::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "l")) ")" )))))) ; theorem :: AMISTD_1:20 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "st" (Bool (Bool (Set ($#k2_compos_0 :::"InsCode"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "s")) ")" ) ($#r8_pboole :::"="::: ) (Set ($#k9_memstr_0 :::"IncIC"::: ) "(" (Set (Var "s")) "," (Num 1) ")" ))))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_hidden :::"PartState":::) "of" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Const "N")) ")" ); cluster (Set ($#k6_memstr_0 :::"DataPart"::: ) "p") -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: AMISTD_1:21 (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"preProgram":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v5_amistd_1 :::"really-closed"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set (Var "P")) "," (Set (Var "s")) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))))) ")" )))) ; registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#~v2_compos_1 "non" ($#v2_compos_1 :::"halt-free"::: ) ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v7_amistd_1 :::"parahalting"::: ) for ($#m1_hidden :::"set"::: ) ; end;