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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcndprobprob 24701* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  B  e.  dom  P  /\  ( P `  B )  =/=  0 )  ->  (
 a  e.  dom  P  |->  ( (cprob `  P ) `  <. a ,  B >. ) )  e. Prob )
 
Theorembayesth 24702 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  A )  =/=  0  /\  ( P `  B )  =/=  0 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( ( (cprob `  P ) `  <. B ,  A >. )  x.  ( P `
  A ) ) 
 /  ( P `  B ) ) )
 
19.3.15.3  Real Valued Random Variables
 
Syntaxcrrv 24703 Extend class notation with the class of real valued random variables.
 class rRndVar
 
Definitiondf-rrv 24704 In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
 |- rRndVar  =  ( p  e. Prob  |->  ( dom 
 pMblFnM𝔅 ) )
 
Theoremrrvmbfm 24705 A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )
 
Theoremisrrvv 24706* Elementhood to the set of real-valued random variables with respect to the probability  P. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  ( X : U. dom  P --> RR  /\  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P ) ) )
 
Theoremrrvvf 24707 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : U. dom  P --> RR )
 
Theoremrrvfn 24708 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X  Fn  U. dom  P )
 
Theoremrrvdm 24709 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  dom 
 X  =  U. dom  P )
 
Theoremrrvrnss 24710 The range of a random variable as a subset of  RR. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ran 
 X  C_  RR )
 
Theoremrrvf2 24711 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : dom  X --> RR )
 
Theoremrrvdmss 24712 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 dom  P  C_  dom  X )
 
Theoremrrvfinvima 24713* For a real-value random variable  X, any open interval in 
RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P )
 
Theorem0rrv 24714* The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( x  e. 
 U. dom  P  |->  0 )  e.  (rRndVar `  P ) )
 
Theoremrrvadd 24715 The sum of two random variables is a random variable (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  Y  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ( X  o F  +  Y )  e.  (rRndVar `  P ) )
 
Theoremrrvmulc 24716 A random variable multiplied by a constant is a random variable. (Contributed by Thierry Arnoux, 17-Jan-2017.) (Revised by Thierry Arnoux, 22-May-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( X𝑓/𝑐  x.  C )  e.  (rRndVar `  P ) )
 
Theoremrrvsum 24717 An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X : NN --> (rRndVar `  P ) )   &    |-  ( ( ph  /\  N  e.  NN )  ->  S  =  (  seq  1 (  o F  +  ,  X ) `  N ) )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  S  e.  (rRndVar `  P )
 )
 
19.3.15.4  Preimage set mapping operator
 
Syntaxcorvc 24718 Extend class notation to include the preimage set mapping operator.
 classRV/𝑐 R
 
Definitiondf-orvc 24719* Define the preimage set mapping operator. In probability theory, the notation  P ( X  =  A ) denotes the probability that a random variable  X takes the value  A. We introduce here an operator which enables to write this in Metamath as  ( P `  ( XRV/𝑐  _I  A ) ), and keep a similar notation. Because with this notation  ( XRV/𝑐  _I  A ) is a set, we can also apply it to conditional probabilities, like in  ( P `  ( XRV/𝑐  _I  A )  |  ( YRV/𝑐  _I  B ) ) ).

The oRVC operator transforms a relation  R into an operation taking a random variable  X and a constant  C, and returning the preimage through  X of the equivalence class of  C.

The most commonly used relations are: - equality:  { X  =  A } as  ( XRV/𝑐  _I  A ) cf. ideq 5028- elementhood:  { X  e.  A } as  ( XRV/𝑐  _E  A ) cf. epel 4500- less-than:  { X  <_  A } as  ( XRV/𝑐  <_  A )

Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g. for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.)

 |-RV/𝑐 R  =  ( x  e.  { x  |  Fun  x } ,  a  e.  _V  |->  ( `' x " { y  |  y R a } )
 )
 
Theoremorvcval 24720* Value of the preimage mapping operator applied on a given random variable and constant (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A }
 ) )
 
Theoremorvcval2 24721* Another way to express the value of the preimage mapping operator (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  {
 z  e.  dom  X  |  ( X `  z
 ) R A }
 )
 
Theoremelorvc 24722* Elementhood of a preimage (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ( ph  /\  z  e.  dom  X )  ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorvcval4 24723* The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 24720 (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  U. J  |  y R A } )
 )
 
Theoremorvcoel 24724* If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e. 
 U. J  |  y R A }  e.  J )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremorvccel 24725* If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e. 
 U. J  |  y R A }  e.  ( Clsd `  J )
 )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremelorrvc 24726* Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ( ph  /\  z  e.  U. dom  P ) 
 ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorrvcval4 24727* The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  RR  |  y R A } ) )
 
Theoremorrvcoel 24728* If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( topGen `
  ran  (,) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorrvccel 24729* If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( Clsd `  ( topGen `  ran  (,) ) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorvcgteel 24730 Preimage maps produced by the "greater than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐 `' 
 <_  A )  e.  dom  P )
 
19.3.15.5  Distribution Functions
 
Theoremorvcelval 24731 Preimage maps produced by the "elementhood" relation (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
 
Theoremorvcelel 24732 Preimage maps produced by the "elementhood" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  e.  dom  P )
 
Theoremdstrvval 24733* The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( D `  A )  =  ( P `  ( `' X " A ) ) )
 
Theoremdstrvprob 24734* The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 24733) (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   =>    |-  ( ph  ->  D  e. Prob )
 
19.3.15.6  Cumulative Distribution Functions
 
Theoremorvclteel 24735 Preimage maps produced by the "lower than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  e.  dom  P )
 
Theoremdstfrvel 24736 Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  U. dom  P )   &    |-  ( ph  ->  ( X `  B )  <_  A )   =>    |-  ( ph  ->  B  e.  ( XRV/𝑐  <_  A ) )
 
Theoremdstfrvunirn 24737* The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 ran  ( n  e. 
 NN  |->  ( XRV/𝑐  <_  n ) )  = 
 U. dom  P )
 
Theoremorvclteinc 24738 Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  C_  ( XRV/𝑐  <_  B ) )
 
Theoremdstfrvinc 24739* A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( F `  A )  <_  ( F `  B ) )
 
Theoremdstfrvclim1 24740* The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   =>    |-  ( ph  ->  F  ~~>  1 )
 
19.3.15.7  Probabilities - example
 
Theoremcoinfliplem 24741 Division in the extended real numbers can be used for the coin-flip example. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  =  ( ( #  |`  ~P { H ,  T } )𝑓/𝑐 /𝑒  2 )
 
Theoremcoinflipprob 24742 The  P we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  e. Prob
 
Theoremcoinflipspace 24743 The space of our coin-flip probability (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 dom  P  =  ~P { H ,  T }
 
Theoremcoinflipuniv 24744 The universe of our coin-flip probability is  { H ,  T }. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 U. dom  P  =  { H ,  T }
 
Theoremcoinfliprv 24745 The  X we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  X  e.  (rRndVar `  P )
 
Theoremcoinflippv 24746 The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { H } )  =  (
 1  /  2 )
 
Theoremcoinflippvt 24747 The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { T } )  =  (
 1  /  2 )
 
19.3.15.8  Bertrand's Ballot Problem
 
Theoremballotlemoex 24748*  O is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  O  e.  _V
 
Theoremballotlem1 24749* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( # `  O )  =  ( ( M  +  N )  _C  M )
 
Theoremballotlemelo 24750* Elementhood in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N ) )  /\  ( # `  C )  =  M ) )
 
Theoremballotlem2 24751* The probability that the first vote picked in a count is a B (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   =>    |-  ( P `  { c  e.  O  |  -.  1  e.  c } )  =  ( N  /  ( M  +  N ) )
 
Theoremballotlemfval 24752* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( F `  C ) `  J )  =  ( ( # `  (
 ( 1 ... J )  i^i  C ) )  -  ( # `  (
 ( 1 ... J )  \  C ) ) ) )
 
Theoremballotlemfelz 24753*  ( F `  C ) has values in  ZZ. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( F `  C ) `  J )  e. 
 ZZ )
 
Theoremballotlemfp1 24754* If the  J th ballot is for A,  ( F `  C ) goes up 1. If the  J th ballot is for B,  ( F `  C ) goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   =>    |-  ( ph  ->  (
 ( -.  J  e.  C  ->  ( ( F `
  C ) `  J )  =  (
 ( ( F `  C ) `  ( J  -  1 ) )  -  1 ) ) 
 /\  ( J  e.  C  ->  ( ( F `
  C ) `  J )  =  (
 ( ( F `  C ) `  ( J  -  1 ) )  +  1 ) ) ) )
 
Theoremballotlemfc0 24755*  F takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   &    |-  ( ph  ->  E. i  e.  ( 1 ... J ) ( ( F `
  C ) `  i )  <_  0 )   &    |-  ( ph  ->  0  <  ( ( F `  C ) `  J ) )   =>    |-  ( ph  ->  E. k  e.  ( 1 ... J ) ( ( F `
  C ) `  k )  =  0
 )
 
Theoremballotlemfcc 24756*  F takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   &    |-  ( ph  ->  E. i  e.  ( 1 ... J ) 0  <_  (
 ( F `  C ) `  i ) )   &    |-  ( ph  ->  ( ( F `  C ) `  J )  <  0 )   =>    |-  ( ph  ->  E. k  e.  ( 1 ... J ) ( ( F `
  C ) `  k )  =  0
 )
 
Theoremballotlemfmpn 24757*  ( F `  C ) finishes counting at  ( M  -  N ). (Contributed by Thierry Arnoux, 25-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   =>    |-  ( C  e.  O  ->  ( ( F `  C ) `  ( M  +  N )
 )  =  ( M  -  N ) )
 
Theoremballotlemfval0 24758*  ( F `  C ) always starts counting at 0 . (Contributed by Thierry Arnoux, 25-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   =>    |-  ( C  e.  O  ->  ( ( F `  C ) `  0
 )  =  0 )
 
Theoremballotleme 24759* Elements of  E. (Contributed by Thierry Arnoux, 14-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   =>    |-  ( C  e.  E 
 <->  ( C  e.  O  /\  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  C ) `  i ) ) )
 
Theoremballotlemodife 24760* Elements of  ( O  \  E ). (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   =>    |-  ( C  e.  ( O  \  E )  <-> 
 ( C  e.  O  /\  E. i  e.  (
 1 ... ( M  +  N ) ) ( ( F `  C ) `  i )  <_ 
 0 ) )
 
Theoremballotlem4 24761* If the first pick is a vote for B, A is not ahead throughout the count (Contributed by Thierry Arnoux, 25-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   =>    |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  -.  C  e.  E ) )
 
Theoremballotlem5 24762* If A is not ahead throughout, there is a  k where votes are tied. (Contributed by Thierry Arnoux, 1-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   =>    |-  ( C  e.  ( O  \  E )  ->  E. k  e.  (
 1 ... ( M  +  N ) ) ( ( F `  C ) `  k )  =  0 )
 
Theoremballotlemi 24763* Value of  I for a given counting  C. (Contributed by Thierry Arnoux, 1-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( C  e.  ( O  \  E ) 
 ->  ( I `  C )  =  sup ( {
 k  e.  ( 1
 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
 )  =  0 } ,  RR ,  `'  <  ) )
 
Theoremballotlemiex 24764* Properties of  ( I `  C ). (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( C  e.  ( O  \  E ) 
 ->  ( ( I `  C )  e.  (
 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  ( I `  C ) )  =  0 ) )
 
Theoremballotlemi1 24765* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  ->  ( I `  C )  =/=  1 )
 
Theoremballotlemii 24766* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( ( C  e.  ( O  \  E )  /\  1  e.  C )  ->  ( I `  C )  =/=  1 )
 
Theoremballotlemsup 24767* The set of zeroes of  F satisfies the conditions to have a supremum (Contributed by Thierry Arnoux, 1-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( C  e.  ( O  \  E ) 
 ->  E. z  e.  RR  ( A. w  e.  {
 k  e.  ( 1
 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
 )  =  0 }  -.  z `'  <  w 
 /\  A. w  e.  RR  ( w `'  <  z  ->  E. y  e.  {
 k  e.  ( 1
 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
 )  =  0 } w `'  <  y
 ) ) )
 
Theoremballotlemimin 24768*  ( I `  C ) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( C  e.  ( O  \  E ) 
 ->  -.  E. k  e.  ( 1 ... (
 ( I `  C )  -  1 ) ) ( ( F `  C ) `  k
 )  =  0 )
 
Theoremballotlemic 24769* If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  ->  ( I `  C )  e.  C )
 
Theoremballotlem1c 24770* If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   =>    |-  ( ( C  e.  ( O  \  E )  /\  1  e.  C )  ->  -.  ( I `  C )  e.  C )
 
Theoremballotlemsval 24771* Value of  S (Contributed by Thierry Arnoux, 12-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( S `  C )  =  ( i  e.  ( 1 ... ( M  +  N )
 )  |->  if ( i  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  i ) ,  i
 ) ) )
 
Theoremballotlemsv 24772* Value of  S evaluated at  J for a given counting  C. (Contributed by Thierry Arnoux, 12-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) ) ) 
 ->  ( ( S `  C ) `  J )  =  if ( J  <_  ( I `  C ) ,  (
 ( ( I `  C )  +  1
 )  -  J ) ,  J ) )
 
Theoremballotlemsgt1 24773*  S maps values less than  ( I `  C ) to values greater than 1. (Contributed by Thierry Arnoux, 28-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
 1  <  ( ( S `  C ) `  J ) )
 
Theoremballotlemsdom 24774* Domain of  S for a given counting  C. (Contributed by Thierry Arnoux, 12-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) ) ) 
 ->  ( ( S `  C ) `  J )  e.  ( 1 ... ( M  +  N ) ) )
 
Theoremballotlemsel1i 24775* The range  ( 1 ... ( I `  C
) ) is invariant under  ( S `  C ). (Contributed by Thierry Arnoux, 28-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( S `  C ) `  J )  e.  ( 1 ... ( I `  C ) ) )
 
Theoremballotlemsf1o 24776* The defined  S is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( S `  C ) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) ) 
 /\  `' ( S `  C )  =  ( S `  C ) ) )
 
Theoremballotlemsi 24777* The image by  S of the first tie pick is the first pick. (Contributed by Thierry Arnoux, 14-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( S `  C ) `  ( I `  C ) )  =  1 )
 
Theoremballotlemsima 24778* The image by  S of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( S `  C ) " (
 1 ... J ) )  =  ( ( ( S `  C ) `
  J ) ... ( I `  C ) ) )
 
Theoremballotlemieq 24779* If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  D  e.  ( O  \  E )  /\  ( I `  C )  =  ( I `  D ) )  ->  ( S `  C )  =  ( S `  D ) )
 
Theoremballotlemrval 24780* Value of  R. (Contributed by Thierry Arnoux, 14-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
  C ) " C ) )
 
Theoremballotlemscr 24781* The image of  ( R `  C ) by  ( S `  C ) (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( S `  C ) " ( R `  C ) )  =  C )
 
Theoremballotlemrv 24782* Value of  R evaluated at  J. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) ) ) 
 ->  ( J  e.  ( R `  C )  <->  if ( J  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J )  e.  C )
 )
 
Theoremballotlemrv1 24783* Value of  R before the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <_  ( I `  C ) )  ->  ( J  e.  ( R `  C )  <->  ( ( ( I `  C )  +  1 )  -  J )  e.  C ) )
 
Theoremballotlemrv2 24784* Value of  R after the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  ( I `  C )  <  J )  ->  ( J  e.  ( R `  C )  <->  J  e.  C ) )
 
Theoremballotlemro 24785* Range of  R is included in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
 
Theoremballotlemgval 24786* Expand the value of  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V )  =  ( ( # `
  ( V  i^i  U ) )  -  ( # `
  ( V  \  U ) ) ) )
 
Theoremballotlemgun 24787* A property of the defined  .^ operator (Contributed by Thierry Arnoux, 26-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  V  e.  Fin )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ph  ->  ( V  i^i  W )  =  (/) )   =>    |-  ( ph  ->  ( U  .^  ( V  u.  W ) )  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
 
Theoremballotlemfg 24788* Express the value of  ( F `  C
) in terms of  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 0 ... ( M  +  N ) ) ) 
 ->  ( ( F `  C ) `  J )  =  ( C  .^  ( 1 ... J ) ) )
 
Theoremballotlemfrc 24789* Express the value of  ( F `  ( R `  C )
) in terms of the newly defined  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  ( R `  C ) ) `  J )  =  ( C  .^  ( ( ( S `
  C ) `  J ) ... ( I `  C ) ) ) )
 
Theoremballotlemfrci 24790* Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( F `  ( R `  C ) ) `  ( I `
  C ) )  =  0 )
 
Theoremballotlemfrceq 24791* Value of  F for a reverse counting  ( R `  C ). (Contributed by Thierry Arnoux, 27-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  C ) `  (
 ( ( S `  C ) `  J )  -  1 ) )  =  -u ( ( F `
  ( R `  C ) ) `  J ) )
 
Theoremballotlemfrcn0 24792* Value of  F for a reversed counting  ( R `  C ), before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  ( ( F `  ( R `  C ) ) `  J )  =/=  0 )
 
Theoremballotlemrc 24793* Range of  R. (Contributed by Thierry Arnoux, 19-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E ) )
 
Theoremballotlemirc 24794* Applying  R does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( I `  ( R `
  C ) )  =  ( I `  C ) )
 
Theoremballotlemrinv0 24795* Lemma for ballotlemrinv 24796. (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  D  =  ( ( S `  C ) " C ) ) 
 ->  ( D  e.  ( O  \  E )  /\  C  =  ( ( S `  D ) " D ) ) )
 
Theoremballotlemrinv 24796*  R is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  `' R  =  R
 
Theoremballotlem1ri 24797* When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( 1  e.  ( R `  C )  <->  ( I `  C )  e.  C ) )
 
Theoremballotlem7 24798*  R is a bijection between two subsets of  ( O  \  E
): one where a vote for A is picked first, and one where a vote for B is picked first (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( R  |`  { c  e.  ( O  \  E )  |  1  e.  c } ) : {
 c  e.  ( O 
 \  E )  |  1  e.  c } -1-1-onto-> {
 c  e.  ( O 
 \  E )  |  -.  1  e.  c }
 
Theoremballotlem8 24799* There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( # `  { c  e.  ( O  \  E )  |  1  e.  c } )  =  ( # `  { c  e.  ( O  \  E )  |  -.  1  e.  c } )
 
Theoremballotth 24800* Bertrand's ballot problem : the probability that A is ahead throughout the counting. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( P `  E )  =  ( ( M  -  N )  /  ( M  +  N ) )
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