:: MATRIX_1 semantic presentation begin definitionlet "f" be ($#m1_hidden :::"FinSequence":::); attr "f" is :::"tabular"::: means :: MATRIX_1:def 1 (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f"))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )))); end; :: deftheorem defines :::"tabular"::: MATRIX_1:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_matrix_1 :::"tabular"::: ) ) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v1_matrix_1 :::"tabular"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: MATRIX_1:1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "d")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ))) ; theorem :: MATRIX_1:2 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "m")) ($#k2_finseq_2 :::"|->"::: ) (Set "(" (Set (Var "n")) ($#k2_finseq_2 :::"|->"::: ) (Set (Var "x")) ")" )) "is" ($#v1_matrix_1 :::"tabular"::: ) ))) ; theorem :: MATRIX_1:3 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "s")) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) )) ; theorem :: MATRIX_1:4 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "s1")) "," (Set (Var "s2")) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ))) ; theorem :: MATRIX_1:5 (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ) ; theorem :: MATRIX_1:6 (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ) ; theorem :: MATRIX_1:7 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a1")) ($#k12_finseq_1 :::"*>"::: ) ) "," (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a2")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ))) ; theorem :: MATRIX_1:8 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "b1")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "b1")) "," (Set (Var "b2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#v1_matrix_1 :::"tabular"::: ) ))) ; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "D" ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCOP_1() ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV1_PRE_POLY() bbbadV2_PRE_POLY() ($#v1_matrix_1 :::"tabular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "D" ($#k3_finseq_2 :::"*"::: ) ); end; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; mode Matrix of "D" is ($#v1_matrix_1 :::"tabular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "D" ($#k3_finseq_2 :::"*"::: ) ); end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "D" ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCOP_1() ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV1_PRE_POLY() bbbadV2_PRE_POLY() ($#v1_matrix_1 :::"tabular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "D" ($#k3_finseq_2 :::"*"::: ) ); end; theorem :: MATRIX_1:9 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D"))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "s"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )))) ")" ))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "m", "n" be ($#m1_hidden :::"Nat":::); mode :::"Matrix"::: "of" "m" "," "n" "," "D" -> ($#m2_finseq_1 :::"Matrix":::) "of" "D" means :: MATRIX_1:def 2 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) "m") & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) it))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "n") ")" ) ")" ); end; :: deftheorem defines :::"Matrix"::: MATRIX_1:def 2 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "D"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "b4"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ) ")" ) ")" )))); definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); mode Matrix of "n" "," "D" is ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "n" "," "D"; end; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; mode Matrix of "K" is ($#m2_finseq_1 :::"Matrix":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K"); let "n" be ($#m1_hidden :::"Nat":::); mode Matrix of "n" "," "K" is ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "n" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K"); let "m" be ($#m1_hidden :::"Nat":::); mode Matrix of "n" "," "m" "," "K" is ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "m" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K"); end; theorem :: MATRIX_1:10 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "m")) ($#k5_finseq_2 :::"|->"::: ) (Set "(" (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set (Var "a")) ")" )) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "D")))))) ; theorem :: MATRIX_1:11 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 1) "," (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) "," (Set (Var "D"))))) ; theorem :: MATRIX_1:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p1"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 2) "," (Set (Var "n")) "," (Set (Var "D")))))) ; theorem :: MATRIX_1:13 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "m")) "," (Set (Var "D"))))) ; theorem :: MATRIX_1:14 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 1) "," (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "D")))) ; theorem :: MATRIX_1:15 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 1) "," (Set (Var "D"))))) ; theorem :: MATRIX_1:16 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 2) "," (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "D")))) ; theorem :: MATRIX_1:17 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 1) "," (Num 2) "," (Set (Var "D"))))) ; theorem :: MATRIX_1:18 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a1")) ($#k12_finseq_1 :::"*>"::: ) ) "," (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "a2")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 2) "," (Num 1) "," (Set (Var "D"))))) ; theorem :: MATRIX_1:19 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "b1")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "b1")) "," (Set (Var "b2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 2) "," (Set (Var "D"))))) ; definitionlet "M" be ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::); func :::"width"::: "M" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: MATRIX_1:def 3 (Bool "ex" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "M")) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) it) ")" )) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "M") ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )); end; :: deftheorem defines :::"width"::: MATRIX_1:def 3 : (Bool "for" (Set (Var "M")) "being" ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) "iff" (Bool "ex" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "b2"))) ")" )) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) "iff" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" ) ")" ")" ))); theorem :: MATRIX_1:20 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) "," (Set (Var "n")) "," (Set (Var "D"))) "iff" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) ")" )))) ; definitionlet "M" be ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::); func :::"Indices"::: "M" -> ($#m1_hidden :::"set"::: ) equals :: MATRIX_1:def 4 (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_finseq_1 :::"dom"::: ) "M" ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) "M" ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ); end; :: deftheorem defines :::"Indices"::: MATRIX_1:def 4 : (Bool "for" (Set (Var "M")) "being" ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))); definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "i", "j" be ($#m1_hidden :::"Nat":::); assume (Bool (Set ($#k4_tarski :::"["::: ) (Set (Const "i")) "," (Set (Const "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Const "M")))) ; func "M" :::"*"::: "(" "i" "," "j" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" "D" means :: MATRIX_1:def 5 (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set "M" ($#k1_funct_1 :::"."::: ) "i")) & (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) "j")) ")" )); end; :: deftheorem defines :::"*"::: MATRIX_1:def 5 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) "iff" (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" )) ")" ))))); theorem :: MATRIX_1:21 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M1"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; scheme :: MATRIX_1:sch 1 MatrixLambda{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F4( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) "," (Set F1 "(" ")" ) "st" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "i")) "," (Set (Var "j")) ")" )))) proof end; scheme :: MATRIX_1:sch 2 MatrixEx{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) "," (Set F1 "(" ")" ) "st" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool P1[(Set (Var "i")) "," (Set (Var "j")) "," (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )]))) provided (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" ) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set F3 "(" ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool P1[(Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "x"))]))) proof end; theorem :: MATRIX_1:22 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )))) ; theorem :: MATRIX_1:23 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" )))) ; theorem :: MATRIX_1:24 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" )))) ; theorem :: MATRIX_1:25 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" )))) ; theorem :: MATRIX_1:26 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M2"))))))) ; theorem :: MATRIX_1:27 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M1"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_1:28 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "D")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M1"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M1")))))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); func "M" :::"@"::: -> ($#m2_finseq_1 :::"Matrix":::) "of" "D" means :: MATRIX_1:def 6 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) "M")) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) it)) "iff" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M")) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M"))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"@"::: MATRIX_1:def 6 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "b3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b3")))) "iff" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" )) ")" ) ")" ) ")" ))); definitionlet "n" be ($#m1_hidden :::"Nat":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "D")); :: original: :::"@"::: redefine func "M" :::"@"::: -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "D"; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "i" be ($#m1_hidden :::"Nat":::); func :::"Line"::: "(" "M" "," "i" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" means :: MATRIX_1:def 7 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) "M")) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) "M" ")" )))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" "i" "," (Set (Var "j")) ")" )) ")" ) ")" ); func :::"Col"::: "(" "M" "," "i" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" means :: MATRIX_1:def 8 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "M")) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "M"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," "i" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"Line"::: MATRIX_1:def 7 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" )))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))))); :: deftheorem defines :::"Col"::: MATRIX_1:def 8 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k7_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" )) ")" ) ")" ) ")" ))))); definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "i" be ($#m1_hidden :::"Nat":::); :: original: :::"Line"::: redefine func :::"Line"::: "(" "M" "," "i" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" ($#k1_matrix_1 :::"width"::: ) "M" ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "D"); :: original: :::"Col"::: redefine func :::"Col"::: "(" "M" "," "i" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) "M" ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "D"); end; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); func "n" :::"-Matrices_over"::: "K" -> ($#m1_hidden :::"set"::: ) equals :: MATRIX_1:def 9 (Set "n" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "(" "n" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K") ")" )); func :::"0."::: "(" "K" "," "n" ")" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K" equals :: MATRIX_1:def 10 (Set "n" ($#k5_finseq_2 :::"|->"::: ) (Set "(" "n" ($#k5_finseq_2 :::"|->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "K" ")" ) ")" )); func :::"1."::: "(" "K" "," "n" ")" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K" means :: MATRIX_1:def 11 (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) it))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) "K")) ")" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) it)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "K")) ")" ) ")" ); let "A" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"-"::: "A" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K" means :: MATRIX_1:def 12 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "A"))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" "A" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))); let "B" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func "A" :::"+"::: "B" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K" means :: MATRIX_1:def 13 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "A"))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" "A" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" "B" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))); end; :: deftheorem defines :::"-Matrices_over"::: MATRIX_1:def 9 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "n")) ($#k10_matrix_1 :::"-Matrices_over"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "(" (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K"))) ")" ))))); :: deftheorem defines :::"0."::: MATRIX_1:def 10 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set "(" (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" ) ")" ))))); :: deftheorem defines :::"1."::: MATRIX_1:def 11 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_1 :::"1."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K")))) ")" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b3")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ) ")" )))); :: deftheorem defines :::"-"::: MATRIX_1:def 12 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "," (Set (Var "b4")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k13_matrix_1 :::"-"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) ")" )))); :: deftheorem defines :::"+"::: MATRIX_1:def 13 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "b5")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "B")))) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "B")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) ")" )))); registrationlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); cluster (Set "n" ($#k10_matrix_1 :::"-Matrices_over"::: ) "K") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: MATRIX_1:29 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))) ; theorem :: MATRIX_1:30 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k10_matrix_1 :::"-Matrices_over"::: ) (Set (Var "K")))) "iff" (Bool (Set (Var "x")) "is" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) ")" )))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "K")); attr "M" is :::"diagonal"::: means :: MATRIX_1:def 14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M")) & (Bool (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "K"))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))); end; :: deftheorem defines :::"diagonal"::: MATRIX_1:def 14 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrix_1 :::"diagonal"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) ")" ))); registrationlet "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K") ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCOP_1() ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV1_PRE_POLY() bbbadV2_PRE_POLY() ($#v1_matrix_1 :::"tabular"::: ) ($#v2_matrix_1 :::"diagonal"::: ) for ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "n" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K"); end; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); mode Diagonal of "n" "," "K" is ($#v2_matrix_1 :::"diagonal"::: ) ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K"; end; theorem :: MATRIX_1:31 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "F")) "holds" (Bool (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "A"))))))) ; theorem :: MATRIX_1:32 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "F")) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "B")) ")" ) ($#k14_matrix_1 :::"+"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set "(" (Set (Var "B")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "C")) ")" )))))) ; theorem :: MATRIX_1:33 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "F")) "holds" (Bool (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set "(" ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "F")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A")))))) ; theorem :: MATRIX_1:34 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "F")) "holds" (Bool (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set "(" ($#k13_matrix_1 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "F")) "," (Set (Var "n")) ")" ))))) ; definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); func "n" :::"-G_Matrix_over"::: "F" -> ($#v8_algstr_0 :::"strict"::: ) ($#l2_algstr_0 :::"AbGroup":::) means :: MATRIX_1:def 15 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "n" ($#k10_matrix_1 :::"-Matrices_over"::: ) "F")) & (Bool "(" "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "F" "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "B")))) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k11_matrix_1 :::"0."::: ) "(" "F" "," "n" ")" )) ")" ); end; :: deftheorem defines :::"-G_Matrix_over"::: MATRIX_1:def 15 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b3")) "being" ($#v8_algstr_0 :::"strict"::: ) ($#l2_algstr_0 :::"AbGroup":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k15_matrix_1 :::"-G_Matrix_over"::: ) (Set (Var "F")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k10_matrix_1 :::"-Matrices_over"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "F")) "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b3"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k14_matrix_1 :::"+"::: ) (Set (Var "B")))) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k11_matrix_1 :::"0."::: ) "(" (Set (Var "F")) "," (Set (Var "n")) ")" )) ")" ) ")" )))); theorem :: MATRIX_1:35 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "m")) "," (Set (Var "K")) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))))) ; begin theorem :: MATRIX_1:36 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))))) ; theorem :: MATRIX_1:37 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))))) ; theorem :: MATRIX_1:38 (Bool "for" (Set (Var "M")) "being" ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) ")" ))) ; theorem :: MATRIX_1:39 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) ")" )))) ; definitionlet "F" be ($#m1_hidden :::"Function":::); func :::"Values"::: "F" -> ($#m1_hidden :::"set"::: ) equals :: MATRIX_1:def 16 (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k3_funct_6 :::"rngs"::: ) "F" ")" )); end; :: deftheorem defines :::"Values"::: MATRIX_1:def 16 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k3_funct_6 :::"rngs"::: ) (Set (Var "F")) ")" )))); theorem :: MATRIX_1:40 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) "holds" (Bool (Set (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))))) ; theorem :: MATRIX_1:41 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) "holds" (Bool (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) where f "is" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) : (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "M")))) "}" )))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "D")) ($#k3_finseq_2 :::"*"::: ) ); cluster (Set ($#k16_matrix_1 :::"Values"::: ) "M") -> ($#v1_finset_1 :::"finite"::: ) ; end; theorem :: MATRIX_1:42 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))))))) ; theorem :: MATRIX_1:43 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))))))) ; theorem :: MATRIX_1:44 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: MATRIX_1:45 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) where i, j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) "}" ))) ; theorem :: MATRIX_1:46 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k16_matrix_1 :::"Values"::: ) (Set (Var "M")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ))))) ; theorem :: MATRIX_1:47 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "M"))))))) ;