:: GFACIRC1 semantic presentation begin scheme :: GFACIRC1:sch 1 1AryBooleEx{ F1( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )))) proof end; scheme :: GFACIRC1:sch 2 1AryBooleUniq{ F1( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) } : (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )) ")" )) "holds" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2")))) proof end; scheme :: GFACIRC1:sch 3 1AryBooleDef{ F1( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) } : (Bool "(" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )))) & (Bool "(" "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "x")) ")" )) ")" )) "holds" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2"))) ")" ) ")" ) proof end; definitionfunc :::"inv1"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: GFACIRC1:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"inv1"::: GFACIRC1:def 1 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k1_gfacirc1 :::"inv1"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set (Var "x"))))) ")" )); theorem :: GFACIRC1:1 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set (Var "x")))) & (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_twoscomp :::"nand2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Num 1) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; definitionfunc :::"buf1"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: GFACIRC1:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x")))); end; :: deftheorem defines :::"buf1"::: GFACIRC1:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_gfacirc1 :::"buf1"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ")" )); theorem :: GFACIRC1:2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k2_gfacirc1 :::"buf1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set ($#k2_gfacirc1 :::"buf1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k2_gfacirc1 :::"buf1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k2_gfacirc1 :::"buf1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Num 1) ($#k10_binarith :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ; definitionfunc :::"and2c"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: GFACIRC1:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" )))); end; :: deftheorem defines :::"and2c"::: GFACIRC1:def 3 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k3_gfacirc1 :::"and2c"::: ) )) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" )))) ")" )); theorem :: GFACIRC1:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" ))) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_twoscomp :::"nor2a"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Num 1) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; definitionfunc :::"xor2c"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: GFACIRC1:def 4 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" )))); end; :: deftheorem defines :::"xor2c"::: GFACIRC1:def 4 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k4_gfacirc1 :::"xor2c"::: ) )) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" )))) ")" )); theorem :: GFACIRC1:4 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" ))) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k15_twoscomp :::"xor2a"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k8_twoscomp :::"or2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set "(" (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Num 1) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ; theorem :: GFACIRC1:5 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k15_twoscomp :::"xor2a"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "x")) ($#k10_binarith :::"*>"::: ) ) ")" ) "," (Set "(" (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_binarith :::"<*"::: ) (Set (Var "y")) ($#k10_binarith :::"*>"::: ) ) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) ")" )) ; theorem :: GFACIRC1:6 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )))) ; theorem :: GFACIRC1:7 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "x")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "y")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "y")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "z")) ")" )))) ; theorem :: GFACIRC1:8 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k15_twoscomp :::"xor2a"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )))) ; theorem :: GFACIRC1:9 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set ($#k1_gfacirc1 :::"inv1"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k16_twoscomp :::"xor2b"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) )))) ; begin definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0CarryIStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 5 (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA0CarryIStr"::: GFACIRC1:def 5 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0CarryICirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 6 (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "x" "," "y" "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "y" "," "z" "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "z" "," "x" "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA0CarryICirc"::: GFACIRC1:def 6 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k6_gfacirc1 :::"GFA0CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "z")) "," (Set (Var "x")) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0CarryStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 7 (Set (Set "(" ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA0CarryStr"::: GFACIRC1:def 7 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0CarryCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 8 (Set (Set "(" ($#k6_gfacirc1 :::"GFA0CarryICirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA0CarryCirc"::: GFACIRC1:def 8 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_gfacirc1 :::"GFA0CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 9 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"GFA0CarryOutput"::: GFACIRC1:def 9 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ))); theorem :: GFACIRC1:10 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:11 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:12 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:13 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k5_gfacirc1 :::"GFA0CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:14 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:15 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:16 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:17 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:18 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:19 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:20 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")))) ")" )))) ; theorem :: GFACIRC1:21 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k1_binarith :::"'or'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set (Var "a3"))))))) ; theorem :: GFACIRC1:22 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")))) ")" )))) ; theorem :: GFACIRC1:23 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0AdderStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 10 (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA0AdderStr"::: GFACIRC1:def 10 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0AdderCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 11 (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA0AdderCirc"::: GFACIRC1:def 11 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA0AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 12 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA0AdderOutput"::: GFACIRC1:def 12 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); theorem :: GFACIRC1:24 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:25 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:26 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:27 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:28 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:29 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1a2")) "," (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1a2")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3"))))))) ; theorem :: GFACIRC1:30 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA0Str"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 13 (Set (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA0Str"::: GFACIRC1:def 13 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_gfacirc1 :::"GFA0AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k7_gfacirc1 :::"GFA0CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA0Circ"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 14 (Set (Set "(" ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA0Circ"::: GFACIRC1:def 14 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k14_gfacirc1 :::"BitGFA0Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_gfacirc1 :::"GFA0AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k8_gfacirc1 :::"GFA0CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); theorem :: GFACIRC1:31 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:32 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:33 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:34 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:35 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:36 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:37 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:38 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA0CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 15 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"BitGFA0CarryOutput"::: GFACIRC1:def 15 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k15_gfacirc1 :::"BitGFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA0AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k13_gfacirc1 :::"BitGFA0Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 16 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"BitGFA0AdderOutput"::: GFACIRC1:def 16 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k16_gfacirc1 :::"BitGFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); theorem :: GFACIRC1:39 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k14_gfacirc1 :::"BitGFA0Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k12_gfacirc1 :::"GFA0AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k9_gfacirc1 :::"GFA0CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:40 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k14_gfacirc1 :::"BitGFA0Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; begin definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1CarryIStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 17 (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA1CarryIStr"::: GFACIRC1:def 17 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1CarryICirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 18 (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "x" "," "y" "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "y" "," "z" "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "z" "," "x" "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA1CarryICirc"::: GFACIRC1:def 18 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k18_gfacirc1 :::"GFA1CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "z")) "," (Set (Var "x")) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1CarryStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 19 (Set (Set "(" ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA1CarryStr"::: GFACIRC1:def 19 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1CarryCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 20 (Set (Set "(" ($#k18_gfacirc1 :::"GFA1CarryICirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA1CarryCirc"::: GFACIRC1:def 20 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_gfacirc1 :::"GFA1CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 21 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"GFA1CarryOutput"::: GFACIRC1:def 21 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ))); theorem :: GFACIRC1:41 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:42 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:43 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:44 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k17_gfacirc1 :::"GFA1CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:45 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:46 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:47 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:48 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:49 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:50 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:51 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")))) ")" )))) ; theorem :: GFACIRC1:52 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k1_binarith :::"'or'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set (Var "a3"))))))) ; theorem :: GFACIRC1:53 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")))) ")" )))) ; theorem :: GFACIRC1:54 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1AdderStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 22 (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA1AdderStr"::: GFACIRC1:def 22 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1AdderCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 23 (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA1AdderCirc"::: GFACIRC1:def 23 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA1AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 24 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA1AdderOutput"::: GFACIRC1:def 24 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); theorem :: GFACIRC1:55 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:56 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:57 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:58 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:59 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:60 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1a2")) "," (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1a2")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" )))))) ; theorem :: GFACIRC1:61 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:62 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3")) ")" )))))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA1Str"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 25 (Set (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA1Str"::: GFACIRC1:def 25 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k22_gfacirc1 :::"GFA1AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k19_gfacirc1 :::"GFA1CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA1Circ"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 26 (Set (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA1Circ"::: GFACIRC1:def 26 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k26_gfacirc1 :::"BitGFA1Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k23_gfacirc1 :::"GFA1AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k20_gfacirc1 :::"GFA1CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); theorem :: GFACIRC1:63 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:64 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:65 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:66 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:67 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:68 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:69 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:70 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA1CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 27 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"BitGFA1CarryOutput"::: GFACIRC1:def 27 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k27_gfacirc1 :::"BitGFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k25_twoscomp :::"or3"::: ) ) ($#k4_tarski :::"]"::: ) ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA1AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k25_gfacirc1 :::"BitGFA1Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 28 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"BitGFA1AdderOutput"::: GFACIRC1:def 28 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k28_gfacirc1 :::"BitGFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); theorem :: GFACIRC1:71 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k26_gfacirc1 :::"BitGFA1Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k24_gfacirc1 :::"GFA1AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k21_gfacirc1 :::"GFA1CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a1")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a3")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a3")) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:72 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k2_twoscomp :::"and2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k26_gfacirc1 :::"BitGFA1Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; begin definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2CarryIStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 29 (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA2CarryIStr"::: GFACIRC1:def 29 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2CarryICirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 30 (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "x" "," "y" "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "y" "," "z" "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "z" "," "x" "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA2CarryICirc"::: GFACIRC1:def 30 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k30_gfacirc1 :::"GFA2CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "z")) "," (Set (Var "x")) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2CarryStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 31 (Set (Set "(" ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA2CarryStr"::: GFACIRC1:def 31 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2CarryCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 32 (Set (Set "(" ($#k30_gfacirc1 :::"GFA2CarryICirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA2CarryCirc"::: GFACIRC1:def 32 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k30_gfacirc1 :::"GFA2CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 33 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"GFA2CarryOutput"::: GFACIRC1:def 33 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ))); theorem :: GFACIRC1:73 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:74 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:75 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:76 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k29_gfacirc1 :::"GFA2CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:77 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:78 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:79 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:80 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:81 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:82 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:83 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:84 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set (Var "a1")) ($#k1_binarith :::"'or'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set (Var "a3")) ")" )))))) ; theorem :: GFACIRC1:85 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ")" ) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:86 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2AdderStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 34 (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA2AdderStr"::: GFACIRC1:def 34 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2AdderCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 35 (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA2AdderCirc"::: GFACIRC1:def 35 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA2AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 36 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"GFA2AdderOutput"::: GFACIRC1:def 36 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); theorem :: GFACIRC1:87 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:88 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:89 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:90 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:91 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:92 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1a2")) "," (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1a2")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" )))))) ; theorem :: GFACIRC1:93 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:94 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" )))))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA2Str"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 37 (Set (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA2Str"::: GFACIRC1:def 37 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k34_gfacirc1 :::"GFA2AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k31_gfacirc1 :::"GFA2CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA2Circ"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 38 (Set (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA2Circ"::: GFACIRC1:def 38 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k38_gfacirc1 :::"BitGFA2Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k35_gfacirc1 :::"GFA2AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k32_gfacirc1 :::"GFA2CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); theorem :: GFACIRC1:95 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:96 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:97 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:98 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:99 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:100 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:101 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:102 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA2CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 39 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"BitGFA2CarryOutput"::: GFACIRC1:def 39 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k39_gfacirc1 :::"BitGFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA2AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k37_gfacirc1 :::"BitGFA2Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 40 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ); end; :: deftheorem defines :::"BitGFA2AdderOutput"::: GFACIRC1:def 40 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k40_gfacirc1 :::"BitGFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ")" ))); theorem :: GFACIRC1:103 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k38_gfacirc1 :::"BitGFA2Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k36_gfacirc1 :::"GFA2AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k33_gfacirc1 :::"GFA2CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set (Var "a2")) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ")" ) ")" ))) ")" )))) ; theorem :: GFACIRC1:104 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_gfacirc1 :::"xor2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_gfacirc1 :::"and2c"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k3_twoscomp :::"and2a"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k38_gfacirc1 :::"BitGFA2Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; begin definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3CarryIStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 41 (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA3CarryIStr"::: GFACIRC1:def 41 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3CarryICirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 42 (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "x" "," "y" "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "y" "," "z" "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" "z" "," "x" "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA3CarryICirc"::: GFACIRC1:def 42 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k42_gfacirc1 :::"GFA3CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" ) ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k6_facirc_1 :::"1GateCircuit"::: ) "(" (Set (Var "z")) "," (Set (Var "x")) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3CarryStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 43 (Set (Set "(" ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA3CarryStr"::: GFACIRC1:def 43 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k5_circcomb :::"1GateCircStr"::: ) "(" (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3CarryCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 44 (Set (Set "(" ($#k42_gfacirc1 :::"GFA3CarryICirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )); end; :: deftheorem defines :::"GFA3CarryCirc"::: GFACIRC1:def 44 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k42_gfacirc1 :::"GFA3CarryICirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k7_facirc_1 :::"1GateCircuit"::: ) "(" (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 45 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"GFA3CarryOutput"::: GFACIRC1:def 45 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ))); theorem :: GFACIRC1:105 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:106 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:107 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:108 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k41_gfacirc1 :::"GFA3CarryIStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:109 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:110 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:111 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:112 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:113 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:114 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:115 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:116 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set (Var "a1")) ($#k1_binarith :::"'or'"::: ) (Set (Var "a2")) ")" ) ($#k1_binarith :::"'or'"::: ) (Set (Var "a3")) ")" )))))) ; theorem :: GFACIRC1:117 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ")" ) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ))) ")" )))) ; theorem :: GFACIRC1:118 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3AdderStr"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 46 (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA3AdderStr"::: GFACIRC1:def 46 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_facirc_1 :::"2GatesCircStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3AdderCirc"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 47 (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA3AdderCirc"::: GFACIRC1:def 47 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_facirc_1 :::"2GatesCircuit"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"GFA3AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 48 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"GFA3AdderOutput"::: GFACIRC1:def 48 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); theorem :: GFACIRC1:119 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:120 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:121 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:122 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:123 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:124 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1a2")) "," (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) & (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k6_circuit2 :::"Following"::: ) (Set (Var "s")) ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1a2")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3"))))))) ; theorem :: GFACIRC1:125 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a3")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a1")) ($#k2_binarith :::"'xor'"::: ) (Set (Var "a2")))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a1"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a2"))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Var "a3"))) ")" )))) ; theorem :: GFACIRC1:126 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_twoscomp :::"."::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" )))))) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA3Str"::: "(" "x" "," "y" "," "z" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v11_struct_0 "non" ($#v11_struct_0 :::"void"::: ) ) ($#v1_msualg_1 :::"strict"::: ) ($#v1_circcomb :::"unsplit"::: ) ($#v2_circcomb :::"gate`1=arity"::: ) ($#v3_circcomb :::"gate`2isBoolean"::: ) ($#l1_msualg_1 :::"ManySortedSign"::: ) equals :: GFACIRC1:def 49 (Set (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA3Str"::: GFACIRC1:def 49 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k46_gfacirc1 :::"GFA3AdderStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k2_circcomb :::"+*"::: ) (Set "(" ($#k43_gfacirc1 :::"GFA3CarryStr"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA3Circ"::: "(" "x" "," "y" "," "z" ")" -> ($#v3_msualg_1 :::"strict"::: ) ($#v4_circcomb :::"gate`2=den"::: ) ($#v6_circcomb :::"Boolean"::: ) ($#l3_msualg_1 :::"Circuit":::) "of" (Set ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" "x" "," "y" "," "z" ")" ) equals :: GFACIRC1:def 50 (Set (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" "x" "," "y" "," "z" ")" ")" )); end; :: deftheorem defines :::"BitGFA3Circ"::: GFACIRC1:def 50 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k50_gfacirc1 :::"BitGFA3Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k47_gfacirc1 :::"GFA3AdderCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k3_circcomb :::"+*"::: ) (Set "(" ($#k44_gfacirc1 :::"GFA3CarryCirc"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )))); theorem :: GFACIRC1:127 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k1_enumset1 :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GFACIRC1:128 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#m1_hidden :::"Relation":::))) ; theorem :: GFACIRC1:129 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:130 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) ))) ; theorem :: GFACIRC1:131 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#~v1_xtuple_0 "non" ($#v1_xtuple_0 :::"pair"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) "is" ($#v1_facirc_1 :::"without_pairs"::: ) )) ; theorem :: GFACIRC1:132 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:133 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; theorem :: GFACIRC1:134 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k2_msafree2 :::"InputVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))) ")" )) ; definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA3CarryOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 51 (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "y" "," "z" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "z" "," "x" ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ); end; :: deftheorem defines :::"BitGFA3CarryOutput"::: GFACIRC1:def 51 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k51_gfacirc1 :::"BitGFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ) ($#k11_finseq_1 :::"*>"::: ) ) "," (Set ($#k29_twoscomp :::"nor3"::: ) ) ($#k4_tarski :::"]"::: ) ))); definitionlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; func :::"BitGFA3AdderOutput"::: "(" "x" "," "y" "," "z" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_msafree2 :::"InnerVertices"::: ) (Set "(" ($#k49_gfacirc1 :::"BitGFA3Str"::: ) "(" "x" "," "y" "," "z" ")" ")" )) equals :: GFACIRC1:def 52 (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" "x" "," "y" "," "z" "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ); end; :: deftheorem defines :::"BitGFA3AdderOutput"::: GFACIRC1:def 52 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k52_gfacirc1 :::"BitGFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_facirc_1 :::"2GatesCircOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ")" ))); theorem :: GFACIRC1:135 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k50_gfacirc1 :::"BitGFA3Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "a3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k48_gfacirc1 :::"GFA3AdderOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" ))) & (Bool (Set (Set "(" ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k45_gfacirc1 :::"GFA3CarryOutput"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_margrel1 :::"'not'"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a2")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ")" ) ")" ) ($#k1_binarith :::"'or'"::: ) (Set "(" (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a3")) ")" ) ($#k10_margrel1 :::"'&'"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set (Var "a1")) ")" ) ")" ) ")" ))) ")" )))) ; theorem :: GFACIRC1:136 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k14_twoscomp :::"xor2"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k4_twoscomp :::"and2b"::: ) ) ($#k4_tarski :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"State":::) "of" (Set "(" ($#k50_gfacirc1 :::"BitGFA3Circ"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ")" ")" ) "holds" (Bool (Set ($#k5_facirc_1 :::"Following"::: ) "(" (Set (Var "s")) "," (Num 2) ")" ) "is" ($#v1_circuit2 :::"stable"::: ) ))) ;