Precondition: A[0..n-1] is an array of numbers, n (element of) N.
Postcondition: Returns the maximum subsequence sum of A.
MaxSum3(A[0..n-1])
	if n = 0
		return 0
	else
		return Max(MaxSum3(A[0..n-2]),MaxSuffix3(A[0..n-1]))


Precondition: A[0..n-1] is an array of numbers, n (element of) N.
Postcondition: Returns the maximum suffix sum of A.
MaxSuffix3(A[0..n-1])
	if n = 0
		return 0
	else
		return Max(0,A[n - 1] + MaxSuffix3(A[0..n - 2]))
		
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Claim A: "MAXSUFFIX3" meets its specifications

Proof: By induction on "n"

Base: n=0
	Then 0 is returned being that 0 itself is the max suffix
	
Induction Hypothesis:
	Assume that n > 0
	if we have an "i" that i < n
	then MAXSUFFIX3(A[0...i-1]) 
	replacing i for n meets the specifications

Induction Step:
	We have already assumed that n > 0 therefore we move on to the else. 
	
	By the Induction Hypothesis we can conclude that...
		MAXSUM3(A[0...i-1]) meets the specifications and by the same logic
		MAXSUM3(A[0...n-2)) does as well since we state
			i < n, and A[0...i-1], thus yeilding "n-2"
		and will then return the maximum suffix sum. We shall call 
		this value "m". This know implies that "else" in MaxSuffix3 can
		be references as
		
			Max(0,A[n \u2212 1] + MaxSuffix3(A[0..n \u2212 2]))
		becomes...
			Max(0,A[n - 1] + m)
		
Claim B: "MAXSUM3" meets its specifications

Proof: by induction on "n"

Base: n=0
	Then 0 is returned being that 0 is the max subsequence sum
	
Induction Hypothesis:
	Assume that n > 0
	if we have an "i" that i < n
	then MAXSUFFIX3(A[0...i-1]) 
	replacing i for n meets the specifications
		
Induction Step: 
	We have already assumed that n > 0 therefore we move on to the else. 
	
	let "sub_value" be equal to what is returned when the maximum 
	subsequence sum is ran using A[0...i-1]
	
	let "sufx_value" be equal to what is returned when the maximum
	suffix sum is ran using A[0...i-1]
	
	By the Induction Hypothesis we can conclude that...
		MAXSUM3(A[0...i-1]) meets the specifications and by the same logic
		MAXSUM3(A[0...n-2)) does as well since we state
			i < n, and A[0...i-1], thus yeilding "n-2"
	
	thus by the decreases value by referencing "i" we say that 
	sub_value^n-1 is returned.
	
	**I'm using ^ to denote a subscript and not a superscript.**
	
	and by the same logic sufx_value^n-1 is returned