:: AMI_WSTD semantic presentation begin 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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "l1", "l2" be ($#m1_hidden :::"Nat":::); pred "l1" :::"<="::: "l2" "," "S" means :: AMI_WSTD:def 1 (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 :::"="::: ) "l1") & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) "l2") & (Bool "(" "for" (Set (Var "n")) "being" ($#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")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) "," "S" ")" )) ")" ) ")" )); end; :: deftheorem defines :::"<="::: AMI_WSTD:def 1 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," (Set (Var "S"))) "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 "l1"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "l2"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#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")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) "," (Set (Var "S")) ")" )) ")" ) ")" )) ")" )))); theorem :: AMI_WSTD:1 (Bool "for" (Set (Var "l3")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "N")) "being" ($#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 "l1")) "," (Set (Var "l2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," (Set (Var "S"))) & (Bool (Set (Var "l2")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l3")) "," (Set (Var "S")))) "holds" (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l3")) "," (Set (Var "S"))))))) ; 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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); attr "S" is :::"InsLoc-antisymmetric"::: means :: AMI_WSTD:def 2 (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," "S") & (Bool (Set (Var "l2")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l1")) "," "S")) "holds" (Bool (Set (Var "l1")) ($#r1_hidden :::"="::: ) (Set (Var "l2")))); end; :: deftheorem defines :::"InsLoc-antisymmetric"::: AMI_WSTD: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"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v1_ami_wstd :::"InsLoc-antisymmetric"::: ) ) "iff" (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," (Set (Var "S"))) & (Bool (Set (Var "l2")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l1")) "," (Set (Var "S")))) "holds" (Bool (Set (Var "l1")) ($#r1_hidden :::"="::: ) (Set (Var "l2")))) ")" ))); definitionlet "N" be ($#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 :::"weakly_standard"::: means :: AMI_WSTD:def 3 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," "S") ")" ) ")" ) ")" )); end; :: deftheorem defines :::"weakly_standard"::: AMI_WSTD: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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v2_ami_wstd :::"weakly_standard"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "S"))) ")" ) ")" ) ")" )) ")" ))); theorem :: AMI_WSTD: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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "S"))) ")" ) ")" ) & (Bool (Set (Var "f2")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "S"))) ")" ) ")" )) "holds" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2")))))) ; theorem :: AMI_WSTD:3 (Bool "for" (Set (Var "N")) "being" ($#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 "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "S"))) ")" ) ")" ) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "k")) ")" ) "," (Set (Var "S")) ")" )) & (Bool "(" "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "k")) ")" ) "," (Set (Var "S")) ")" ))) "holds" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) ")" ) ")" )) ")" )))) ; theorem :: AMI_WSTD: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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v2_ami_wstd :::"weakly_standard"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "k")) ")" ) "," (Set (Var "S")) ")" )) & (Bool "(" "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "k")) ")" ) "," (Set (Var "S")) ")" ))) "holds" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) ")" ) ")" ) ")" ) ")" )) ")" ))) ; begin registrationlet "N" be ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_amistd_1 :::"STC"::: ) "N") -> ($#v2_ami_wstd :::"weakly_standard"::: ) ; 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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) 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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "k" be ($#m1_hidden :::"Nat":::); func :::"il."::: "(" "S" "," "k" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: AMI_WSTD:def 4 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," "S") ")" ) ")" ) & (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) "k")) ")" )); end; :: deftheorem defines :::"il."::: AMI_WSTD: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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set (Var "k")) ")" )) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_ami_wstd :::"<="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "S"))) ")" ) ")" ) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )) ")" ))))); theorem :: AMI_WSTD:5 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "k1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "k2")) ")" ))) "holds" (Bool (Set (Var "k1")) ($#r1_hidden :::"="::: ) (Set (Var "k2")))))) ; theorem :: AMI_WSTD:6 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "k")) ")" )))))) ; 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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "l" be ($#m1_hidden :::"Nat":::); func :::"locnum"::: "(" "l" "," "S" ")" -> ($#m1_hidden :::"Nat":::) means :: AMI_WSTD:def 5 (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" "S" "," it ")" ) ($#r1_hidden :::"="::: ) "l"); end; :: deftheorem defines :::"locnum"::: AMI_WSTD: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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "," (Set (Var "b4")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k2_ami_wstd :::"locnum"::: ) "(" (Set (Var "l")) "," (Set (Var "S")) ")" )) "iff" (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set (Var "b4")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "l"))) ")" )))); definitionlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "l" be ($#m1_hidden :::"Nat":::); :: original: :::"locnum"::: redefine func :::"locnum"::: "(" "l" "," "S" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: AMI_WSTD:7 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l1")) "," (Set (Var "T")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l2")) "," (Set (Var "T")) ")" ))) "holds" (Bool (Set (Var "l1")) ($#r1_hidden :::"="::: ) (Set (Var "l2")))))) ; theorem :: AMI_WSTD:8 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "k1")) ")" ) ($#r1_ami_wstd :::"<="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "k2")) ")" ) "," (Set (Var "T"))) "iff" (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) ")" )))) ; theorem :: AMI_WSTD:9 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l1")) "," (Set (Var "T")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l2")) "," (Set (Var "T")) ")" )) "iff" (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," (Set (Var "T"))) ")" )))) ; theorem :: AMI_WSTD:10 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool (Set (Var "T")) "is" ($#v1_ami_wstd :::"InsLoc-antisymmetric"::: ) ))) ; 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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v1_ami_wstd :::"InsLoc-antisymmetric"::: ) 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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "f" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m1_hidden :::"Nat":::); func "f" :::"+"::: "(" "k" "," "S" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMI_WSTD:def 6 (Set ($#k1_ami_wstd :::"il."::: ) "(" "S" "," (Set "(" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" "f" "," "S" ")" ")" ) ($#k2_nat_1 :::"+"::: ) "k" ")" ) ")" ); end; :: deftheorem defines :::"+"::: AMI_WSTD:def 6 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "f")) ($#k4_ami_wstd :::"+"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set "(" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ) ")" )))))); theorem :: AMI_WSTD:11 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k4_ami_wstd :::"+"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "T")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))))) ; theorem :: AMI_WSTD:12 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k4_ami_wstd :::"+"::: ) "(" (Set (Var "z")) "," (Set (Var "T")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k4_ami_wstd :::"+"::: ) "(" (Set (Var "z")) "," (Set (Var "T")) ")" ))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))))) ; theorem :: AMI_WSTD:13 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k3_ami_wstd :::"locnum"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_ami_wstd :::"+"::: ) "(" (Set (Var "z")) "," (Set (Var "T")) ")" ")" ) "," (Set (Var "T")) ")" )))))) ; definitionlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "f" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"NextLoc"::: "(" "f" "," "S" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMI_WSTD:def 7 (Set "f" ($#k4_ami_wstd :::"+"::: ) "(" (Num 1) "," "S" ")" ); end; :: deftheorem defines :::"NextLoc"::: AMI_WSTD:def 7 : (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k4_ami_wstd :::"+"::: ) "(" (Num 1) "," (Set (Var "S")) ")" ))))); theorem :: AMI_WSTD:14 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set "(" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: AMI_WSTD:15 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"<>"::: ) (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ))))) ; theorem :: AMI_WSTD:16 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "g")) "," (Set (Var "T")) ")" ))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))))) ; theorem :: AMI_WSTD:17 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "z"))))) ; theorem :: AMI_WSTD:18 (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 ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "s")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" ) "," (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) ")" ))))) ; theorem :: AMI_WSTD:19 (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")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "l")) "," (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) ")" ")" ) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: AMI_WSTD:20 (Bool "for" (Set (Var "N")) "being" ($#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 ($#k7_domain_1 :::"{"::: ) (Set (Var "l")) "," (Set "(" ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "l")) "," (Set "(" ($#k4_amistd_1 :::"STC"::: ) (Set (Var "N")) ")" ) ")" ")" ) ($#k7_domain_1 :::"}"::: ) )))) ; 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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "i" be ($#m1_subset_1 :::"Instruction":::) "of" (Set (Const "S")); attr "i" is :::"sequential"::: means :: AMI_WSTD:def 8 (Bool "for" (Set (Var "s")) "being" ($#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 ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" ) "," "S" ")" ))); end; :: deftheorem defines :::"sequential"::: AMI_WSTD: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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "i")) "is" ($#v3_ami_wstd :::"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"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set "(" ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s")) ")" ) "," (Set (Var "S")) ")" ))) ")" )))); theorem :: AMI_WSTD:21 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "il")) "being" ($#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" ($#v3_ami_wstd :::"sequential"::: ) )) "holds" (Bool (Set ($#k1_amistd_1 :::"NIC"::: ) "(" (Set (Var "i")) "," (Set (Var "il")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_ami_wstd :::"NextLoc"::: ) "(" (Set (Var "il")) "," (Set (Var "S")) ")" ")" ) ($#k6_domain_1 :::"}"::: ) )))))) ; theorem :: AMI_WSTD:22 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "i")) "is" ($#v3_ami_wstd :::"sequential"::: ) )) "holds" (Bool "not" (Bool (Set (Var "i")) "is" ($#v2_extpro_1 :::"halting"::: ) ))))) ; registrationlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); cluster ($#v3_ami_wstd :::"sequential"::: ) -> ($#~v2_extpro_1 "non" ($#v2_extpro_1 :::"halting"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S"); cluster ($#v2_extpro_1 :::"halting"::: ) -> ($#~v3_ami_wstd "non" ($#v3_ami_wstd :::"sequential"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S"); end; begin definitionlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "F" be (Set ($#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 :::"para-closed"::: means :: AMI_WSTD:def 9 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" "S" "st" (Bool (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" "S" "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" "F" "," (Set (Var "s")) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F")))); end; :: deftheorem defines :::"para-closed"::: AMI_WSTD:def 9 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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"))) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_ami_wstd :::"para-closed"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) "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 :: AMI_WSTD:23 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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" ($#v5_amistd_1 :::"really-closed"::: ) ) & (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "F")) "is" ($#v4_ami_wstd :::"para-closed"::: ) )))) ; theorem :: AMI_WSTD:24 (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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) "holds" (Bool (Set (Set "(" ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" ($#k2_compos_1 :::"halt"::: ) (Set (Var "S")) ")" )) "is" ($#v5_amistd_1 :::"really-closed"::: ) ))) ; 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"))) ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::); attr "F" is :::"lower"::: means :: AMI_WSTD:def 10 (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F"))) "holds" (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l")) "," "S")) "holds" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F")))); end; :: deftheorem defines :::"lower"::: AMI_WSTD: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"))) ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v5_ami_wstd :::"lower"::: ) ) "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 "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l")) "," (Set (Var "S")))) "holds" (Bool (Set (Var "m")) ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#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 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) -> (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: AMI_WSTD:25 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set (Set "(" ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "i"))) "is" ($#v5_ami_wstd :::"lower"::: ) )))) ; registrationlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) (Num 1) ($#v3_card_1 :::"-element"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_ami_wstd :::"lower"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: AMI_WSTD:26 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))))) ; theorem :: AMI_WSTD:27 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#v1_setfam_1 :::"with_zero"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_memstr_0 :::"with_non-empty_values"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "P")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b3"))) ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P")))) "iff" (Bool (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "P")))) ")" ))))) ; 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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::); func :::"LastLoc"::: "F" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: AMI_WSTD:def 11 (Bool "ex" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v6_membered :::"natural-membered"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l")) "," "S" ")" ")" ) where l "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F")) "}" ) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" "S" "," (Set "(" ($#k1_xxreal_2 :::"max"::: ) (Set (Var "M")) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"LastLoc"::: AMI_WSTD: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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F")))) "iff" (Bool "ex" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v6_membered :::"natural-membered"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "l")) "," (Set (Var "S")) ")" ")" ) where l "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")))) "}" ) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set "(" ($#k1_xxreal_2 :::"max"::: ) (Set (Var "M")) ")" ) ")" )) ")" )) ")" ))))); theorem :: AMI_WSTD:28 (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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))))) ; theorem :: AMI_WSTD:29 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#v5_relat_1 :::"-valued"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) "holds" (Bool (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F"))) ($#r1_ami_wstd :::"<="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "G"))) "," (Set (Var "T")))))) ; theorem :: AMI_WSTD:30 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "l")) ($#r1_ami_wstd :::"<="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F"))) "," (Set (Var "T"))))))) ; theorem :: AMI_WSTD:31 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "G")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set (Var "b2"))) ($#v5_relat_1 :::"-valued"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Var "G"))))))) ; theorem :: AMI_WSTD:32 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "T")) "," (Set "(" (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "F")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))))) ; registrationlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_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"::: ) ($#v5_amistd_1 :::"really-closed"::: ) ($#v5_ami_wstd :::"lower"::: ) -> (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v4_ami_wstd :::"para-closed"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::); attr "F" is :::"halt-ending"::: means :: AMI_WSTD:def 12 (Bool (Set "F" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k6_ami_wstd :::"LastLoc"::: ) "F" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_compos_1 :::"halt"::: ) "S")); attr "F" is :::"unique-halt"::: means :: AMI_WSTD:def 13 (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set "F" ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_compos_1 :::"halt"::: ) "S")) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "F"))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) "F"))); end; :: deftheorem defines :::"halt-ending"::: AMI_WSTD:def 12 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v6_ami_wstd :::"halt-ending"::: ) ) "iff" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k6_ami_wstd :::"LastLoc"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_compos_1 :::"halt"::: ) (Set (Var "S")))) ")" )))); :: deftheorem defines :::"unique-halt"::: AMI_WSTD:def 13 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v7_ami_wstd :::"unique-halt"::: ) ) "iff" (Bool "for" (Set (Var "f")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_compos_1 :::"halt"::: ) (Set (Var "S")))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k6_ami_wstd :::"LastLoc"::: ) (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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_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"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_ami_wstd :::"lower"::: ) ($#v6_ami_wstd :::"halt-ending"::: ) ($#v7_ami_wstd :::"unique-halt"::: ) for ($#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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_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"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) ($#v5_ami_wstd :::"lower"::: ) 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"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#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"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#v6_ami_wstd :::"halt-ending"::: ) ($#v7_ami_wstd :::"unique-halt"::: ) for ($#m1_hidden :::"set"::: ) ; 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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); mode pre-Macro of "S" is (Set ($#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_finset_1 :::"finite"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#v6_ami_wstd :::"halt-ending"::: ) ($#v7_ami_wstd :::"unique-halt"::: ) ($#m1_hidden :::"Function":::); 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"::: ) ($#v3_extpro_1 :::"halting"::: ) ($#v2_ami_wstd :::"weakly_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"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v5_amistd_1 :::"really-closed"::: ) ($#v5_ami_wstd :::"lower"::: ) ($#v6_ami_wstd :::"halt-ending"::: ) ($#v7_ami_wstd :::"unique-halt"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: AMI_WSTD:33 (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"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_amistd_1 :::"SUCC"::: ) "(" (Set (Var "l1")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "holds" (Bool (Set (Var "l1")) ($#r1_ami_wstd :::"<="::: ) (Set (Var "l2")) "," (Set (Var "S")))))) ; definitionlet "N" be ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Const "N")); let "loc" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m1_hidden :::"Nat":::); func "loc" :::"-'"::: "(" "k" "," "S" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: AMI_WSTD:def 14 (Set ($#k1_ami_wstd :::"il."::: ) "(" "S" "," (Set "(" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" "loc" "," "S" ")" ")" ) ($#k7_nat_d :::"-'"::: ) "k" ")" ) ")" ); end; :: deftheorem defines :::"-'"::: AMI_WSTD:def 14 : (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "loc")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "loc")) ($#k7_ami_wstd :::"-'"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_ami_wstd :::"il."::: ) "(" (Set (Var "S")) "," (Set "(" (Set "(" ($#k3_ami_wstd :::"locnum"::: ) "(" (Set (Var "loc")) "," (Set (Var "S")) ")" ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k")) ")" ) ")" )))))); theorem :: AMI_WSTD:34 (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "l")) ($#k7_ami_wstd :::"-'"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "l")))))) ; theorem :: AMI_WSTD:35 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#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"::: ) ($#v3_memstr_0 :::"IC-Ins-separated"::: ) ($#v2_ami_wstd :::"weakly_standard"::: ) ($#l1_extpro_1 :::"AMI-Struct"::: ) "over" (Set (Var "N")) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "l")) ($#k4_ami_wstd :::"+"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" ")" ) ($#k7_ami_wstd :::"-'"::: ) "(" (Set (Var "k")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "l"))))))) ;