:: SCMFSA7B semantic presentation begin theorem :: SCMFSA7B:1 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set ($#k2_compos_1 :::"halt"::: ) (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )))) "implies" (Bool (Set (Set "(" ($#k2_scmfsa6a :::"Directed"::: ) (Set "(" ($#k11_compos_1 :::"Macro"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k11_scmfsa_2 :::"goto"::: ) (Num 2))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_compos_1 :::"halt"::: ) (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )))) "implies" (Bool (Set (Set "(" ($#k2_scmfsa6a :::"Directed"::: ) (Set "(" ($#k11_compos_1 :::"Macro"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "i"))) ")" ")" )) ; theorem :: SCMFSA7B:2 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_scmfsa6a :::"Directed"::: ) (Set "(" ($#k11_compos_1 :::"Macro"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k11_scmfsa_2 :::"goto"::: ) (Num 2)))) ; registrationlet "a" be ($#m1_subset_1 :::"Int-Location":::); let "k" be ($#m1_hidden :::"Integer":::); cluster (Set "a" ($#k1_scmfsa_7 :::":="::: ) "k") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_afinsq_1 :::"initial"::: ) ; end; registrationlet "a" be ($#m1_subset_1 :::"Int-Location":::); let "k" be ($#m1_hidden :::"Integer":::); cluster (Set "a" ($#k1_scmfsa_7 :::":="::: ) "k") -> ($#v7_amistd_1 :::"parahalting"::: ) ; end; theorem :: SCMFSA7B:3 (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#v1_scmfsa_m :::"read-write"::: ) ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "a")) ($#k1_scmfsa_7 :::":="::: ) (Set (Var "k")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#v1_scmfsa_m :::"read-write"::: ) ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set (Var "a")))) "holds" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "a")) ($#k1_scmfsa_7 :::":="::: ) (Set (Var "k")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "b")))) ")" ) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "a")) ($#k1_scmfsa_7 :::":="::: ) (Set (Var "k")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k18_scmfsa_2 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k18_scmfsa_2 :::"."::: ) (Set (Var "f")))) ")" ) ")" ))))) ; registrationlet "f" be ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) ; let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); cluster (Set "f" ($#k4_scmfsa_7 :::":="::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_afinsq_1 :::"initial"::: ) ; end; registrationlet "f" be ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) ; let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); cluster (Set "f" ($#k4_scmfsa_7 :::":="::: ) "p") -> ($#v7_amistd_1 :::"parahalting"::: ) ; end; theorem :: SCMFSA7B:4 (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_scmfsa_7 :::":="::: ) (Set (Var "p")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k18_scmfsa_2 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool "(" "for" (Set (Var "a")) "being" ($#v1_scmfsa_m :::"read-write"::: ) ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_scmfsa_2 :::"intloc"::: ) (Num 1))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_scmfsa_2 :::"intloc"::: ) (Num 2)))) "holds" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_scmfsa_7 :::":="::: ) (Set (Var "p")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")))) ")" ) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"<>"::: ) (Set (Var "f")))) "holds" (Bool (Set (Set "(" ($#k1_scmfsa6b :::"IExec"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_scmfsa_7 :::":="::: ) (Set (Var "p")) ")" ) "," (Set (Var "P")) "," (Set (Var "s")) ")" ")" ) ($#k18_scmfsa_2 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k18_scmfsa_2 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))))) ; definitionlet "i" be ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); let "a" be ($#m1_subset_1 :::"Int-Location":::); pred "i" :::"refers"::: "a" means :: SCMFSA7B:def 1 (Bool "not" (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "b")) ($#k6_scmfsa_2 :::":="::: ) "a") ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "b")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "b")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k9_scmfsa_2 :::"MultBy"::: ) "(" (Set (Var "b")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "b")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" "a" "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "a" ($#k12_scmfsa_2 :::"=0_goto"::: ) (Set (Var "l"))) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "a" ($#k13_scmfsa_2 :::">0_goto"::: ) (Set (Var "l"))) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set (Set (Var "b")) ($#k14_scmfsa_2 :::":="::: ) "(" (Set (Var "f")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "(" (Set (Var "f")) "," (Set (Var "b")) ")" ($#k15_scmfsa_2 :::":="::: ) "a") ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "(" (Set (Var "f")) "," "a" ")" ($#k15_scmfsa_2 :::":="::: ) (Set (Var "b"))) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set (Set (Var "f")) ($#k17_scmfsa_2 :::":=<0,...,0>"::: ) "a") ($#r1_hidden :::"<>"::: ) "i") ")" ))))); end; :: deftheorem defines :::"refers"::: SCMFSA7B:def 1 : (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_scmfsa7b :::"refers"::: ) (Set (Var "a"))) "iff" (Bool "not" (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "b")) ($#k6_scmfsa_2 :::":="::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k9_scmfsa_2 :::"MultBy"::: ) "(" (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "a")) ($#k12_scmfsa_2 :::"=0_goto"::: ) (Set (Var "l"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "a")) ($#k13_scmfsa_2 :::">0_goto"::: ) (Set (Var "l"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "b")) ($#k14_scmfsa_2 :::":="::: ) "(" (Set (Var "f")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set "(" (Set (Var "f")) "," (Set (Var "b")) ")" ($#k15_scmfsa_2 :::":="::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set "(" (Set (Var "f")) "," (Set (Var "a")) ")" ($#k15_scmfsa_2 :::":="::: ) (Set (Var "b"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "f")) ($#k17_scmfsa_2 :::":=<0,...,0>"::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) ")" ))))) ")" ))); definitionlet "I" be ($#m1_hidden :::"preProgram":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); let "a" be ($#m1_subset_1 :::"Int-Location":::); pred "I" :::"refers"::: "a" means :: SCMFSA7B:def 2 (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "I")) & (Bool (Set (Var "i")) ($#r1_scmfsa7b :::"refers"::: ) "a") ")" )); end; :: deftheorem defines :::"refers"::: SCMFSA7B:def 2 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"preProgram":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r2_scmfsa7b :::"refers"::: ) (Set (Var "a"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "I")))) & (Bool (Set (Var "i")) ($#r1_scmfsa7b :::"refers"::: ) (Set (Var "a"))) ")" )) ")" ))); definitionlet "i" be ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); let "a" be ($#m1_subset_1 :::"Int-Location":::); pred "i" :::"destroys"::: "a" means :: SCMFSA7B:def 3 (Bool "not" (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool "(" (Bool (Set "a" ($#k6_scmfsa_2 :::":="::: ) (Set (Var "b"))) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k7_scmfsa_2 :::"AddTo"::: ) "(" "a" "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" "a" "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k9_scmfsa_2 :::"MultBy"::: ) "(" "a" "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" "a" "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "b")) "," "a" ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "a" ($#k14_scmfsa_2 :::":="::: ) "(" (Set (Var "f")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) "i") & (Bool (Set "a" ($#k16_scmfsa_2 :::":=len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) "i") ")" )))); end; :: deftheorem defines :::"destroys"::: SCMFSA7B:def 3 : (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))) "iff" (Bool "not" (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k6_scmfsa_2 :::":="::: ) (Set (Var "b"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k9_scmfsa_2 :::"MultBy"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "a")) ($#k14_scmfsa_2 :::":="::: ) "(" (Set (Var "f")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "a")) ($#k16_scmfsa_2 :::":=len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) ")" )))) ")" ))); definitionlet "I" be (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); let "a" be ($#m1_subset_1 :::"Int-Location":::); pred "I" :::"destroys"::: "a" means :: SCMFSA7B:def 4 (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "I")) & (Bool (Set (Var "i")) ($#r3_scmfsa7b :::"destroys"::: ) "a") ")" )); end; :: deftheorem defines :::"destroys"::: SCMFSA7B:def 4 : (Bool "for" (Set (Var "I")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r4_scmfsa7b :::"destroys"::: ) (Set (Var "a"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "I")))) & (Bool (Set (Var "i")) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))) ")" )) ")" ))); definitionlet "I" be (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); attr "I" is :::"good"::: means :: SCMFSA7B:def 5 (Bool (Bool "not" "I" ($#r4_scmfsa7b :::"destroys"::: ) (Set ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) )))); end; :: deftheorem defines :::"good"::: SCMFSA7B:def 5 : (Bool "for" (Set (Var "I")) "being" (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_scmfsa7b :::"good"::: ) ) "iff" (Bool (Bool "not" (Set (Var "I")) ($#r4_scmfsa7b :::"destroys"::: ) (Set ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ")" )); theorem :: SCMFSA7B:5 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "holds" (Bool (Bool "not" (Set ($#k2_compos_1 :::"halt"::: ) (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:6 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set (Set (Var "b")) ($#k6_scmfsa_2 :::":="::: ) (Set (Var "c"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:7 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "b")) "," (Set (Var "c")) ")" ) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:8 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "b")) "," (Set (Var "c")) ")" ) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:9 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set ($#k9_scmfsa_2 :::"MultBy"::: ) "(" (Set (Var "b")) "," (Set (Var "c")) ")" ) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:10 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "c")))) "holds" (Bool "not" (Bool (Set ($#k10_scmfsa_2 :::"Divide"::: ) "(" (Set (Var "b")) "," (Set (Var "c")) ")" ) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) ; theorem :: SCMFSA7B:11 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Bool "not" (Set ($#k11_scmfsa_2 :::"goto"::: ) (Set (Var "l"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:12 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Bool "not" (Set (Set (Var "b")) ($#k12_scmfsa_2 :::"=0_goto"::: ) (Set (Var "l"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:13 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "l")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Bool "not" (Set (Set (Var "b")) ($#k13_scmfsa_2 :::">0_goto"::: ) (Set (Var "l"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:14 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set (Set (Var "b")) ($#k14_scmfsa_2 :::":="::: ) "(" (Set (Var "f")) "," (Set (Var "c")) ")" ) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:15 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool (Bool "not" (Set "(" (Set (Var "f")) "," (Set (Var "c")) ")" ($#k15_scmfsa_2 :::":="::: ) (Set (Var "b"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:16 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set (Set (Var "b")) ($#k16_scmfsa_2 :::":=len"::: ) (Set (Var "f"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; theorem :: SCMFSA7B:17 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "f")) "being" ($#m1_scmfsa_2 :::"FinSeq-Location"::: ) "holds" (Bool (Bool "not" (Set (Set (Var "f")) ($#k17_scmfsa_2 :::":=<0,...,0>"::: ) (Set (Var "b"))) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a")))))) ; definitionlet "I" be ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); let "s" be ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); let "P" be ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ); pred "I" :::"is_closed_on"::: "s" "," "P" means :: SCMFSA7B:def 6 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set "(" "P" ($#k1_funct_4 :::"+*"::: ) "I" ")" ) "," (Set "(" ($#k8_memstr_0 :::"Initialize"::: ) "s" ")" ) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) "I"))); pred "I" :::"is_halting_on"::: "s" "," "P" means :: SCMFSA7B:def 7 (Bool (Set "P" ($#k1_funct_4 :::"+*"::: ) "I") ($#r1_extpro_1 :::"halts_on"::: ) (Set ($#k8_memstr_0 :::"Initialize"::: ) "s")); end; :: deftheorem defines :::"is_closed_on"::: SCMFSA7B:def 6 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r5_scmfsa7b :::"is_closed_on"::: ) (Set (Var "s")) "," (Set (Var "P"))) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_memstr_0 :::"IC"::: ) (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set "(" (Set (Var "P")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k8_memstr_0 :::"Initialize"::: ) (Set (Var "s")) ")" ) "," (Set (Var "k")) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "I"))))) ")" )))); :: deftheorem defines :::"is_halting_on"::: SCMFSA7B:def 7 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r6_scmfsa7b :::"is_halting_on"::: ) (Set (Var "s")) "," (Set (Var "P"))) "iff" (Bool (Set (Set (Var "P")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "I"))) ($#r1_extpro_1 :::"halts_on"::: ) (Set ($#k8_memstr_0 :::"Initialize"::: ) (Set (Var "s")))) ")" )))); theorem :: SCMFSA7B:18 (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v6_amistd_1 :::"paraclosed"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool (Set (Var "I")) ($#r5_scmfsa7b :::"is_closed_on"::: ) (Set (Var "s")) "," (Set (Var "P"))))) ")" )) ; theorem :: SCMFSA7B:19 (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v7_amistd_1 :::"parahalting"::: ) ) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "holds" (Bool (Set (Var "I")) ($#r6_scmfsa7b :::"is_halting_on"::: ) (Set (Var "s")) "," (Set (Var "P"))))) ")" )) ; theorem :: SCMFSA7B:20 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Instruction":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r3_scmfsa7b :::"destroys"::: ) (Set (Var "a"))))) "holds" (Bool (Set (Set "(" ($#k2_extpro_1 :::"Exec"::: ) "(" (Set (Var "i")) "," (Set (Var "s")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))))) ; theorem :: SCMFSA7B:21 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"State":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"Instruction-Sequence":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Program":::) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) "st" (Bool (Bool (Bool "not" (Set (Var "I")) ($#r4_scmfsa7b :::"destroys"::: ) (Set (Var "a")))) & (Bool (Set (Var "I")) ($#r5_scmfsa7b :::"is_closed_on"::: ) (Set (Var "s")) "," (Set (Var "P")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_extpro_1 :::"Comput"::: ) "(" (Set "(" (Set (Var "P")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k8_memstr_0 :::"Initialize"::: ) (Set (Var "s")) ")" ) "," (Set (Var "k")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))))))) ; registration cluster (Set ($#k4_compos_1 :::"Stop"::: ) (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) -> ($#v7_amistd_1 :::"parahalting"::: ) ($#v1_scmfsa7b :::"good"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v5_ordinal1 :::"T-Sequence-like"::: ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_afinsq_1 :::"initial"::: ) ($#v7_amistd_1 :::"parahalting"::: ) ($#v1_scmfsa7b :::"good"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_compos_1 :::"InstructionsF"::: ) "of" (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) )) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_afinsq_1 :::"initial"::: ) ($#v6_amistd_1 :::"paraclosed"::: ) ($#v1_scmfsa7b :::"good"::: ) -> ($#v1_scmfsa6b :::"keeping_0"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: SCMFSA7B:22 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_scmfsa_7 :::"aSeq"::: ) "(" (Set (Var "a")) "," (Set (Var "k")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set "(" (Set (Var "a")) ($#k6_scmfsa_2 :::":="::: ) (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ) "," (Set "(" ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "a")) "," (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ")" ) "," (Set "(" ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "a")) "," (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ")" ) ($#k1_enumset1 :::"}"::: ) )))) ; theorem :: SCMFSA7B:23 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Int-Location":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "a")) ($#k1_scmfsa_7 :::":="::: ) (Set (Var "k")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_enumset1 :::"{"::: ) (Set "(" ($#k2_compos_1 :::"halt"::: ) (Set ($#k1_scmfsa_2 :::"SCM+FSA"::: ) ) ")" ) "," (Set "(" (Set (Var "a")) ($#k6_scmfsa_2 :::":="::: ) (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ) "," (Set "(" ($#k7_scmfsa_2 :::"AddTo"::: ) "(" (Set (Var "a")) "," (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ")" ) "," (Set "(" ($#k8_scmfsa_2 :::"SubFrom"::: ) "(" (Set (Var "a")) "," (Set "(" ($#k4_scmfsa_2 :::"intloc"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ")" ")" ) ($#k2_enumset1 :::"}"::: ) )))) ; registrationlet "a" be ($#v1_scmfsa_m :::"read-write"::: ) ($#m1_subset_1 :::"Int-Location":::); let "k" be ($#m1_hidden :::"Integer":::); cluster (Set "a" ($#k1_scmfsa_7 :::":="::: ) "k") -> ($#v1_scmfsa7b :::"good"::: ) ; end; registrationlet "a" be ($#v1_scmfsa_m :::"read-write"::: ) ($#m1_subset_1 :::"Int-Location":::); let "k" be ($#m1_hidden :::"Integer":::); cluster (Set "a" ($#k1_scmfsa_7 :::":="::: ) "k") -> ($#v1_scmfsa6b :::"keeping_0"::: ) ; end;