(* This pulls on the following posts:
- https://blog.relevant.community/bonding-curves-in-depth-intuition-parametrization-d3905a681e0a
- https://medium.com/@aventus/token-bonding-curves-547f3a04914
- 
*)

dS[dB_,S0_,B0_,RR_]:=S0 * ((1 + dB / B0) ^ (RR) - 1)
dB[dS_,S0_,B0_,RR_]:= B0 * (1 - (1 - dS / S0)^(1/RR))
RR := 1/3
s0 := 1
b0 := 1
b1 := 2
s1 := s0+dS[b1-b0,s0,b0,RR]
If[
dB[s1-s0,s1,b1,RR] == b0,
"SUCCESS: Buying 1 BitClout worth of creator coin and then selling works", 
"ERROR: Buying 1 BitClout worth of creator coin and selling *DOES NOT* work"
]

In[1]:= (*Someone buys db0 worth of creator coin*)
vs1 := s_start+dS[db0,s_start,b_start,RR]
vb1 := b_start+db0
ds0  := vs1-s_start
(*We add subsidy to the pool without modifying anything else*)
vb1 = vb1+subsidy;
(*Now the user sells their initial amount and gets some new bitclout*)
dbFinal := dB[ds0,vs1,vb1,RR]
Simplify[dbFinal];
(*Can dbFinal ever be more than db0+subsidy?*)
Print["The diff below should be >0 for all possible values of subsidy, db0, and b_start:"]
Print["db0+subsidy - dbFinal = ", Simplify[db0+subsidy-dbFinal]]
(*Nice properties of the quantity that results from the above:
- As long as subsidy, b_start, and db0 are all >0 then this quantity should always be >0 as desired.
- If subsidy is zero then the diff is zero. This makes sense because what you put in (=db0) should equal what you get out (=dbFinal) when there isn't a subsidy.
- If b_start is small then this quantity will be small, meaning the amount the user loses due to "slippage" from our fudging factor should also be small. *)
(*Follow-up: Is there a way to make it so that the slippage from this bullshit is zero?*)

(*Working out formulas for the polynomial version to make them match Bancor*)
polyPriceWithSlope[m_,s_,RR_]:=m*s^(1/RR-1)
polyBitcloutWithSlope[m_,s_,RR_]:=m/(1/RR)*s^(1/RR)
polyDb[dS_,m_,s_,RR_]:=polyBitcloutWithSlope[m,s+dS,RR]-polyBitcloutWithSlope[m,s,RR]
(*This formula assumes you know b, the amount of BitClout purchased*)
simplePolyDb[dS_,m_,s_,b_,RR_]:= b*((dS/s+1)^(1/RR)-1)
(*The simplified formula should equal the full formula*)
If[
(Simplify[Simplify[polyDb[kk, mm,ss,rr]]/Simplify[simplePolyDb[kk,mm,ss,polyBitcloutWithSlope[mm,ss,rr], rr]]]/.ss->3) == 1,
Print["SUCCESS: The simplified formula for polynomial dB equals the full formula"],
Print["ERROR: The simplified formula for polynomial dB is *NOT EQUAL* to the full formula"]]
RR=.
dsVar /.Solve[polyDb[dsVar,m,s,RR]==dB,dsVar][[1]]
polyDs[dB_,m_,s_,RR_]:=-s + (-((-dB - m RR s^(1/RR))/(m RR)))^RR 
(*This formula assumes you know b, the amount of BitClout purchased*)
simplePolyDs[db_,m_,s_,b_,RR_]:=s*((db/b+1)^(RR)-1)
If[Simplify[polyDs[dB,m,s,rr]/simplePolyDs[dB,m,s,polyBitcloutWithSlope[m,s,rr],rr] /.{s->3,dB->5,rr->7,m->11}]==1,
Print["SUCCESS: The simplied formula for polynomial dS equals the full formula"],
Print["ERROR: The simplied formula for polynomial dS *DOES NOT* equal the full formula"]]

(*This agrees with what you get when you solve: c0 x^(1/RR-1) = b/(s*RR) *)
slopeInTermsOfVars=.
slopeInTermsOfVars[s_,b_,RR_]:=b*1/RR/s^(1/RR)
If[slopeInTermsOfVars[1,1,1/3.]==3,
Print["SUCCESS: Formula for m in terms of (s,b,RR) is working!"],
Print["ERROR:  Formula for m in terms of (s,b,RR) is NOT working"]]
If[polyDs[1,slopeInTermsOfVars[1,1,1/3.],1,1/3.]==0.2599210498948732`,
"SUCCESS: Polynomial formulas match Bancor formulas!",
"ERROR: Polynomial formulas do *NOT* match Bancor formulas!"]

If[(Simplify[polyDb[s,m,0,RR]]/.RR->1/3)==m*s^3/3,
"SUCCESS: Formula for Db at zero is correct",
"ERROR: Formula for Db at zero is WRONG"]
If[(Simplify[polyDs[p,m,0,RR]]/.RR->1/3)==3^(1/3)*(p/m)^(1/3),
"SUCCESS: Formula for Ds at zero is correct",
"ERROR: Formula for Ds at zero is WRONG"]

In[122]:= 
(*How do our price curves change as BitClout price increases?*)
(**)
(*This is the price in USD given some amount of USD has been put into a profile*)
priceUSDGivenUSDPurchased[usdPurchased_, bitcloutPriceUSD_]:=polyPriceWithSlope[mm,polyDs[usdPurchased/bitcloutPriceUSD,mm,0,rr],rr]
(*Compute the above function for two different values of bitcloutPriceUSD and divide them*)
xxx[usdPurchased_,bitcloutPriceUSD_]:=Simplify[priceUSDGivenUSDPurchased[usdPurchased,bitcloutPriceUSD,mm,rr]/priceUSDGivenUSDPurchased[usdPurchased,bitcloutPriceUSD * p,mm,rr]]
(*The result should be p^1/3*)
If[(Simplify[xxx[usdPurchased,bitcloutPriceUSD]/.{mm->5,rr->1/3,usdPurchased->7,bitcloutPriceUSD->11}]/p^(1/3)/.p->13) == 13^(1/3),
"SUCCESS: The USD price of a profile given a certain amount of USD purchased is scaled by bitcloutPriceUSD^(1/3)",
"ERROR: The USD price of a profile is *NOT* scaled by bitcloutPriceUSD^(1/3)"
]

In[19]:= (*Scratch below this point*)
(*Start with a Bancor pool with some amount of BitClout and zero creator coin*)
bStart = 3;
sStart = 0;
rr=1/3;
bitcloutPutIn = 5;
m = slopeInTermsOfVars[1,1,rr];
(*Buy some coins initially using the polynomial model*)
coinPurchased =polyDs[bitcloutPutIn,m,0,rr];
(*Get the polynomial price after buying some coins using the polynomial model*)
N[polyPriceWithSlope[m,coinPurchased,rr]]
(*Get the Bancor price after buying some coins using the polynomial model*)
N[(bStart+bitcloutPutIn)/(coinPurchased * rr)]
(*Now sell this creator coin for $BitClout using the Bancor formula*)
(*db0 = dB[coinPurchased, sStart+coinPurchased,bStart+bitcloutPutIn,rr]*)
m=.
Print[-s + (-((-dB - m RR s^(1/RR))/(m RR)))^RR /.s->0]