:: DIST_1 semantic presentation begin notationlet "S" be ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); synonym :::"whole_event"::: "s" for :::"dom"::: "S"; end; theorem :: DIST_1:1 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k9_xtuple_0 :::"whole_event"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k8_relset_1 :::"""::: ) (Set (Var "S")))))) ; notationlet "S" be ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); let "x" be ($#m1_hidden :::"set"::: ) ; synonym :::"event_pick"::: "(" "x" "," "s" ")" for :::"Coim"::: "(" "S" "," "s" ")" ; end; definitionlet "S" be ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); let "x" be ($#m1_hidden :::"set"::: ) ; :: original: :::"event_pick"::: redefine func :::"event_pick"::: "(" "x" "," "s" ")" -> ($#m1_subset_1 :::"Event":::) "of" (Set "(" ($#k9_xtuple_0 :::"whole_event"::: ) ")" ); end; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"frequency"::: "(" "x" "," "s" ")" -> ($#m1_hidden :::"Nat":::) equals :: DIST_1:def 1 (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_dist_1 :::"event_pick"::: ) "(" "x" "," "s" ")" ")" )); end; :: deftheorem defines :::"frequency"::: DIST_1:def 1 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_dist_1 :::"frequency"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_dist_1 :::"event_pick"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ")" )))))); theorem :: DIST_1:2 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"whole_event"::: ) ))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set ($#k1_dist_1 :::"event_pick"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" )))))) ; theorem :: DIST_1:3 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k9_xtuple_0 :::"whole_event"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")))))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"FDprobability"::: "(" "x" "," "s" ")" -> ($#m1_subset_1 :::"Real":::) equals :: DIST_1:def 2 (Set (Set "(" ($#k2_dist_1 :::"frequency"::: ) "(" "x" "," "s" ")" ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "s" ")" )); end; :: deftheorem defines :::"FDprobability"::: DIST_1:def 2 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_dist_1 :::"frequency"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" )))))); theorem :: DIST_1:4 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_dist_1 :::"frequency"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ")" )))))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); func :::"FDprobSEQ"::: "s" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: DIST_1:def 3 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) "S" ")" ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set "(" (Set "(" ($#k1_uproots :::"canFS"::: ) "S" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) "," "s" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"FDprobSEQ"::: DIST_1:def 3 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set "(" (Set "(" ($#k1_uproots :::"canFS"::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set (Var "s")) ")" )) ")" ) ")" ) ")" )))); theorem :: DIST_1:5 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rpr_1 :::"prob"::: ) (Set "(" ($#k1_dist_1 :::"event_pick"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ")" )))))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s", "t" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); pred "s" "," "t" :::"-are_prob_equivalent"::: means :: DIST_1:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," "s" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," "t" ")" ))); reflexivity (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" )))) ; symmetry (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "t")) ")" )) ")" )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" )))) ; end; :: deftheorem defines :::"-are_prob_equivalent"::: DIST_1:def 4 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_dist_1 :::"FDprobability"::: ) "(" (Set (Var "x")) "," (Set (Var "t")) ")" ))) ")" ))); theorem :: DIST_1:6 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "u")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) & (Bool (Set (Var "t")) "," (Set (Var "u")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) )) "holds" (Bool (Set (Var "s")) "," (Set (Var "u")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); func :::"Finseq-EQclass"::: "s" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "S" ($#k3_finseq_2 :::"*"::: ) ")" ) equals :: DIST_1:def 5 "{" (Set (Var "t")) where t "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" : (Bool "s" "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "}" ; end; :: deftheorem defines :::"Finseq-EQclass"::: DIST_1:def 5 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) : (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "}" ))); registrationlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); cluster (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) "s") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: DIST_1:7 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "iff" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s")))) ")" ))) ; theorem :: DIST_1:8 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s")))))) ; theorem :: DIST_1:9 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "iff" (Bool (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "t")))) ")" ))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; func :::"distribution_family"::: "S" -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" "S" ($#k3_finseq_2 :::"*"::: ) ")" ) means :: DIST_1:def 6 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "S" ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s"))))) ")" )); end; :: deftheorem defines :::"distribution_family"::: DIST_1:def 6 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" (Set (Var "S")) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "S")) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s"))))) ")" )) ")" ))); registrationlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k6_dist_1 :::"distribution_family"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: DIST_1:10 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ) "iff" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "t")))) ")" ))) ; theorem :: DIST_1:11 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s"))))) "holds" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "t")))))) ; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; func :::"GenProbSEQ"::: "S" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k6_dist_1 :::"distribution_family"::: ) "S" ")" ) "," (Set "(" (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) means :: DIST_1:def 7 (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) "S") (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "x"))) & (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")))) ")" ))); end; :: deftheorem defines :::"GenProbSEQ"::: DIST_1:def 7 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_dist_1 :::"GenProbSEQ"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S"))) (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")))) ")" ))) ")" ))); theorem :: DIST_1:12 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k7_dist_1 :::"GenProbSEQ"::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k5_dist_1 :::"Finseq-EQclass"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")))))) ; registrationlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_dist_1 :::"GenProbSEQ"::: ) "S") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); assume (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")) "st" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Const "p")))) ; func :::"distribution"::: "(" "p" "," "S" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) "S") means :: DIST_1:def 8 (Bool (Set (Set "(" ($#k7_dist_1 :::"GenProbSEQ"::: ) "S" ")" ) ($#k3_funct_2 :::"."::: ) it) ($#r1_hidden :::"="::: ) "p"); end; :: deftheorem defines :::"distribution"::: DIST_1:def 8 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_dist_1 :::"distribution"::: ) "(" (Set (Var "p")) "," (Set (Var "S")) ")" )) "iff" (Bool (Set (Set "(" ($#k7_dist_1 :::"GenProbSEQ"::: ) (Set (Var "S")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )))); definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); func :::"freqSEQ"::: "s" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: DIST_1:def 9 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) "S" ")" ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) "s" ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) "s" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"freqSEQ"::: DIST_1:def 9 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s")))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" )))); theorem :: DIST_1:13 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" )))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set "(" ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_dist_1 :::"frequency"::: ) "(" (Set (Var "x")) "," (Set (Var "s")) ")" )) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_uproots :::"canFS"::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) ")" ))))) ; theorem :: DIST_1:14 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" ) ($#k10_rvsum_1 :::"*"::: ) (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" ))))) ; theorem :: DIST_1:15 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" ) ")" ))))) ; theorem :: DIST_1:16 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "s"))))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set "(" ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k2_dist_1 :::"frequency"::: ) "(" (Set "(" (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set (Var "s")) ")" )) & (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_uproots :::"canFS"::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m")))) ")" ))))) ; theorem :: DIST_1:17 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "L")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "S")) "is" ($#v1_prob_2 :::"disjoint_valued"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "L")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "S"))))) "holds" (Bool "(" (Bool (Set (Set (Var "S")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "S")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "S"))) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "L")))) ")" ))) ; theorem :: DIST_1:18 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k9_dist_1 :::"freqSEQ"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")))))) ; theorem :: DIST_1:19 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)))) ; registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); cluster (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) "s") -> ($#v1_matrprob :::"ProbFinS"::: ) ; end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; mode :::"distProbFinS"::: "of" "S" -> ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: DIST_1:def 10 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) "S")) & (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "st" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) it)) ")" ); end; :: deftheorem defines :::"distProbFinS"::: DIST_1:def 10 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_dist_1 :::"distProbFinS"::: ) "of" (Set (Var "S"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "S")))) & (Bool "ex" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Var "b2")))) ")" ) ")" ))); theorem :: DIST_1:20 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_dist_1 :::"distProbFinS"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set ($#k8_dist_1 :::"distribution"::: ) "(" (Set (Var "p")) "," (Set (Var "S")) ")" ) "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S")))) & (Bool (Set (Set "(" ($#k7_dist_1 :::"GenProbSEQ"::: ) (Set (Var "S")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k8_dist_1 :::"distribution"::: ) "(" (Set (Var "p")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ))) ; begin definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "S")); attr "s" is :::"uniformly_distributed"::: means :: DIST_1:def 11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) "s" ")" )))) "holds" (Bool (Set (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) "s" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) "S" ")" )))); end; :: deftheorem defines :::"uniformly_distributed"::: DIST_1:def 11 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" )))) "holds" (Bool (Set (Set "(" ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" )))) ")" ))); theorem :: DIST_1:21 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) )) "holds" (Bool (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set (Var "s"))) "is" ($#v3_funct_1 :::"constant"::: ) ))) ; theorem :: DIST_1:22 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ) & (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) )) "holds" (Bool (Set (Var "t")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ))) ; theorem :: DIST_1:23 (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ) & (Bool (Set (Var "t")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) )) "holds" (Bool (Set (Var "s")) "," (Set (Var "t")) ($#r1_dist_1 :::"-are_prob_equivalent"::: ) ))) ; registrationlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_uproots :::"canFS"::: ) "S") -> ($#v1_dist_1 :::"uniformly_distributed"::: ) ; end; definitionlet "S" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; func :::"uniform_distribution"::: "S" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) "S") means :: DIST_1:def 12 (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "S" "holds" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ) ")" )); end; :: deftheorem defines :::"uniform_distribution"::: DIST_1:def 12 : (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k6_dist_1 :::"distribution_family"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_dist_1 :::"uniform_distribution"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "s")) "is" ($#v1_dist_1 :::"uniformly_distributed"::: ) ) ")" )) ")" ))); registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) ($#v1_finset_1 :::"finite"::: ) bbbadV1_VALUED_0() bbbadV2_VALUED_0() bbbadV3_VALUED_0() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV4_PARTFUN3() ($#v1_matrprob :::"ProbFinS"::: ) for ($#m1_dist_1 :::"distProbFinS"::: ) "of" "S"; end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; func :::"Uniform_FDprobSEQ"::: "S" -> ($#m1_dist_1 :::"distProbFinS"::: ) "of" "S" equals :: DIST_1:def 13 (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set "(" ($#k1_uproots :::"canFS"::: ) "S" ")" )); end; :: deftheorem defines :::"Uniform_FDprobSEQ"::: DIST_1:def 13 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k11_dist_1 :::"Uniform_FDprobSEQ"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k4_dist_1 :::"FDprobSEQ"::: ) (Set "(" ($#k1_uproots :::"canFS"::: ) (Set (Var "S")) ")" )))); registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_dist_1 :::"Uniform_FDprobSEQ"::: ) "S") -> ($#v3_funct_1 :::"constant"::: ) ; end; theorem :: DIST_1:24 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_dist_1 :::"uniform_distribution"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k8_dist_1 :::"distribution"::: ) "(" (Set "(" ($#k11_dist_1 :::"Uniform_FDprobSEQ"::: ) (Set (Var "S")) ")" ) "," (Set (Var "S")) ")" ))) ;