replaced use of \phi with \emptyset

README.md

@@ -572,14 +572,15 @@ There are other implementations of the surreals. For instance
 + Haskell https://github.com/Lacaranian/surreal (only integers)
 + Haskell https://github.com/serialhex/Surreal-Numbers (not quite working)
 + Haskell https://github.com/elfakyn/Haskell-surreals (only integers)
++ Javascript http://www.iwriteiam.nl/SurrealNumbers.html
 + Mathematica http://demonstrations.wolfram.com/GeneratingTheSurrealNumbers/
               (only generation)
++ Lua https://github.com/Ryan1729/lua-Surreal-numbers?files=1 
 + Python https://github.com/codeinthehole/python-surreal (not much
          implemented)
 + Python https://github.com/314eter/surreal-numbers
 + Ruby http://raganwald.com/2009/03/07/elegance-and-the-surreals.html
 
-
 And some of these languages might be more appropriate in some ways for
 this task. But I wanted to learn Julia, and see how far I could take
 it here. Moreover, most of these are at least as incomplete as the

examples/Data/multiplication_table.tex

@@ -6,12 +6,40 @@
 \text{$\overline{2} \times \overline{2}$} & 6 & 11 & 14 & 5 & 0.0 & 13 & 6 & 11 \\
 \text{$\overline{3} \times \overline{2}$} & 12 & 45 & 82 & 625 & 0.07 & 83 & 9 & 69 \\
 \text{$\overline{4} \times \overline{2}$} & 20 & 153 & 331 & \text{7.5E+05} & 0.01 & 460 & 12 & 393 \\
-\text{$\overline{5} \times \overline{2}$} & 30 & 428 & 1015 & \text{6.1E+09} & 0.12 & 1935 & 15 & 1705 \\
-\text{$\overline{6} \times \overline{2}$} & 42 & 1024 & 2565 & \text{2.9E+14} & 0.74 & 6330 & 18 & 5708 \\
-\text{$\overline{7} \times \overline{2}$} & 56 & 2172 & 5637 & \text{7.0E+19} & 3.9 & 17234 & 21 & 15802 \\
-\text{$\overline{8} \times \overline{2}$} & 72 & 4196 & 11160 & \text{8.6E+25} & 15.94 & 40904 & 24 & 37970 \\
-\text{$\overline{9} \times \overline{2}$} & 90 & 7529 & 20384 & \text{5.0E+32} & 56.7 & 87363 & 27 & 81860 \\
-\text{$\overline{3} \times \overline{3}$} & 31 & 1627 & 6042 & \text{4.3E+12} & 0.35 & 3485 & 10 & 3242 \\
-\text{$\overline{4} \times \overline{2}$} & 42 & 14961 & 80196 & \text{7.3E+19} & 9.08 & 23448 & 30 & 21510 \\
+\text{$\overline{5} \times \overline{2}$} & 30 & 428 & 1015 & \text{6.1E+09} & 0.1 & 1935 & 15 & 1705 \\
+\text{$\overline{6} \times \overline{2}$} & 42 & 1024 & 2565 & \text{2.9E+14} & 0.58 & 6330 & 18 & 5708 \\
+\text{$\overline{7} \times \overline{2}$} & 56 & 2172 & 5637 & \text{7.0E+19} & 2.9 & 17234 & 21 & 15802 \\
+\text{$\overline{8} \times \overline{2}$} & 72 & 4196 & 11160 & \text{8.6E+25} & 11.84 & 40904 & 24 & 37970 \\
+\text{$\overline{9} \times \overline{2}$} & 90 & 7529 & 20384 & \text{5.0E+32} & 40.99 & 87363 & 27 & 81860 \\
+\text{$\overline{10} \times \overline{2}$} & 110 & 12729 & 34928 & \text{1.3E+38} & 129.2 & 171690 & 30 & 162059 \\
+\text{$\overline{11} \times \overline{2}$} & 132 & 20495 & 56828 & \text{ overflow } & 360.0 & 315502 & 33 & 299559 \\
+\text{$\overline{12} \times \overline{2}$} & 156 & 31683 & 88585 & \text{ overflow } & 895.2 & 548628 & 36 & 523415 \\
+\text{$\overline{2} \times \overline{-2}$} & 6 & 11 & 14 & 5 & 0.0 & 13 & 9 & 12 \\
+\text{$\overline{1} \times \overline{3}$} & 3 & 4 & 3 & 1 & 0.0 & 3 & 7 & 3 \\
+\text{$\overline{2} \times \overline{3}$} & 12 & 45 & 82 & 625 & 0.0 & 83 & 9 & 69 \\
+\text{$\overline{3} \times \overline{3}$} & 31 & 737 & 2052 & \text{9.4E+10} & 0.25 & 3419 & 10 & 3182 \\
+\text{$\overline{4} \times \overline{3}$} & 64 & 16958 & 64653 & \text{6.1E+27} & 184.1 & 286358 & 14 & 281775 \\
+\text{$\overline{1} \times \overline{4}$} & 4 & 5 & 4 & 1 & 0.0 & 4 & 9 & 4 \\
+\text{$\overline{2} \times \overline{4}$} & 20 & 153 & 331 & \text{7.5E+05} & 0.01 & 460 & 12 & 393 \\
+\text{$\overline{1} \times \overline{1//2}$} & 2 & 3 & 3 & 2 & 0.0 & 2 & 5 & 2 \\
+\text{$\overline{2} \times \overline{1//2}$} & 6 & 15 & 31 & 29 & 0.0 & 18 & 8 & 15 \\
+\text{$\overline{3} \times \overline{1//2}$} & 12 & 65 & 174 & 5397 & 0.0 & 125 & 11 & 102 \\
+\text{$\overline{4} \times \overline{1//2}$} & 20 & 216 & 650 & \text{8.1E+06} & 0.02 & 684 & 14 & 582 \\
+\text{$\overline{5} \times \overline{1//2}$} & 30 & 582 & 1869 & \text{7.7E+10} & 0.17 & 2740 & 17 & 2413 \\
+\text{$\overline{6} \times \overline{1//2}$} & 42 & 1344 & 4497 & \text{4.0E+15} & 1.25 & 8574 & 20 & 7733 \\
+\text{$\overline{7} \times \overline{1//2}$} & 56 & 2766 & 9520 & \text{1.1E+21} & 8.23 & 22498 & 23 & 20635 \\
+\text{$\overline{8} \times \overline{1//2}$} & 72 & 5211 & 18308 & \text{1.4E+27} & 46.92 & 51816 & 26 & 48113 \\
+\text{$\overline{9} \times \overline{1//2}$} & 90 & 9157 & 32679 & \text{8.9E+33} & 226.3 & 107982 & 29 & 101204 \\
+\text{$\overline{10} \times \overline{1//2}$} & 110 & 15213 & 54963 & \text{ overflow } & 986.0 & 207950 & 32 & 196322 \\
+\text{$\overline{11} \times \overline{1//2}$} & 132 & 24135 & 88066 & \text{1.5E+37} & 3727.0 & 375716 & 35 & 356784 \\
+\text{$\overline{12} \times \overline{1//2}$} & 156 & 36842 & 135534 & \text{9.0E+37} & 12588.0 & 644052 & 38 & 614528 \\
+\text{$\overline{1} \times \overline{1//4}$} & 3 & 4 & 5 & 3 & 0.0 & 3 & 7 & 3 \\
+\text{$\overline{2} \times \overline{1//4}$} & 12 & 108 & 426 & 48218 & 0.0 & 163 & 11 & 128 \\
+\text{$\overline{3} \times \overline{1//4}$} & 31 & 4828 & 30377 & \text{1.9E+16} & 6.21 & 24455 & 15 & 23192 \\
+\text{$\overline{1//2} \times \overline{1//2}$} & 6 & 13 & 27 & 28 & 0.0 & 15 & 6 & 12 \\
+\text{$\overline{1//2} \times \overline{1//4}$} & 12 & 93 & 374 & 37674 & 0.0 & 139 & 9 & 109 \\
+\text{$\overline{1//4} \times \overline{1//4}$} & 31 & 4755 & 34920 & \text{1.6E+17} & 7.39 & 19020 & 10 & 17836 \\
+\text{$\overline{4} \times \overline{2}$} & 42 & 1905 & 5980 & \text{6.7E+15} & 2.92 & 16108 & 30 & 14904 \\
+\text{$\overline{0} \times \overline{2}$} & 6 & 16 & 24 & 20 & 0.0 & 29 & 11 & 24 \\
 \end{array}
 \right]

examples/Data/multiplication_tables.csv

@@ -1,13 +1,41 @@
 "i","n1","n2","generation","nodes","edges","paths","density","time","bytes","adds","prods","created","equals","leq"
-0,0//1,2//1,0,1,0,1.0,1.0,8.019e-6,1216.0,0,1,0,0,0
-1,1//1,2//1,2,3,2,1.0,1.0,7.8116e-5,10912.0,2,5,2,0,0
-2,2//1,2//1,6,11,14,5.0,1.3636363636363635,0.000214553,48608.0,13,6,11,0,0
-3,3//1,2//1,12,45,82,625.0,1.8444444444444446,0.068835036,1.298532e6,83,9,69,0,0
-4,4//1,2//1,20,153,331,748077.0,2.1699346405228757,0.013266295,5.08984e6,460,12,393,0,0
-5,5//1,2//1,30,428,1015,6.085358435e9,2.3738317757009346,0.122446419,4.3640128e7,1935,15,1705,0,0
-6,6//1,2//1,42,1024,2565,2.85343216243798e14,2.505859375,0.743089202,2.71792416e8,6330,18,5708,0,0
-7,7//1,2//1,56,2172,5637,7.0352688036251116e19,2.595764272559853,3.897952511,1.486446016e9,17234,21,15802,0,0
-8,8//1,2//1,72,4196,11160,8.615540969651363e25,2.659914204003813,15.940410716,5.622187952e9,40904,24,37970,0,0
-9,9//1,2//1,90,7529,20384,5.0443458043182515e32,2.7075308805950327,56.699081147,1.6828904896e10,87363,27,81860,0,0
-10,3//1,3//1,31,1627,6042,4.326138913672e12,3.71419791026429,0.353743509,1.48495424e8,3485,10,3242,0,0
-11,4//1,2//1,42,14961,80196,7.27448190950702e19,5.360403716329122,9.077178159,3.59409528e9,23448,30,21510,0,0
+0,0//1,2//1,0,1,0,1.0,1.0,6.522e-6,1216.0,0,1,0,0,0
+1,1//1,2//1,2,3,2,1.0,1.0,8.2687e-5,10912.0,2,5,2,0,0
+2,2//1,2//1,6,11,14,5.0,1.3636363636363635,0.000249978,48096.0,13,6,11,0,0
+3,3//1,2//1,12,45,82,625.0,1.8444444444444446,0.068390523,1.276372e6,83,9,69,0,0
+4,4//1,2//1,20,153,331,748077.0,2.1699346405228757,0.011966549,4.673024e6,460,12,393,0,0
+5,5//1,2//1,30,428,1015,6.085358435e9,2.3738317757009346,0.103476073,3.9358208e7,1935,15,1705,0,0
+6,6//1,2//1,42,1024,2565,2.85343216243798e14,2.505859375,0.575546313,2.37728064e8,6330,18,5708,0,0
+7,7//1,2//1,56,2172,5637,7.0352688036251116e19,2.595764272559853,2.902520881,1.290840368e9,17234,21,15802,0,0
+8,8//1,2//1,72,4196,11160,8.615540969651363e25,2.659914204003813,11.837685148,4.880489248e9,40904,24,37970,0,0
+9,9//1,2//1,90,7529,20384,5.0443458043182515e32,2.7075308805950327,40.991869265,1.3886006016e10,87363,27,81860,0,0
+10,10//1,2//1,110,12729,34928,1.3205472893970104e38,2.744049021918454,129.232637597,4.5452954608e10,171690,30,162059,0,0
+11,11//1,2//1,132,20495,56828,-9.935518445889285e37,2.772822639668212,360.014984408,1.24926111152e11,315502,33,299559,0,0
+12,12//1,2//1,156,31683,88585,-7.282291779914621e37,2.7960104788056688,895.24701852,2.76990644032e11,548628,36,523415,0,0
+13,2//1,-2//1,6,11,14,5.0,1.3636363636363635,0.000283992,55712.0,13,9,12,0,0
+14,1//1,3//1,3,4,3,1.0,1.0,6.3001e-5,16064.0,3,7,3,0,0
+15,2//1,3//1,12,45,82,625.0,1.8444444444444446,0.001342992,459312.0,83,9,69,0,0
+16,3//1,3//1,31,737,2052,9.4415698647e10,2.78561736770692,0.248545979,1.0444824e8,3419,10,3182,0,0
+17,4//1,3//1,64,16958,64653,6.108237799500018e27,3.812595824979361,184.052318177,6.6680640112e10,286358,14,281775,0,0
+18,1//1,4//1,4,5,4,1.0,1.0,8.6929e-5,21216.0,4,9,4,0,0
+19,2//1,4//1,20,153,331,748077.0,2.1699346405228757,0.012298217,4.720544e6,460,12,393,0,0
+20,1//1,1//2,2,3,3,2.0,1.3333333333333333,4.4732e-5,12144.0,2,5,2,0,0
+21,2//1,1//2,6,15,31,29.0,2.1333333333333333,0.000293467,89424.0,18,8,15,0,0
+22,3//1,1//2,12,65,174,5397.0,2.6923076923076925,0.002329558,877584.0,125,11,102,0,0
+23,4//1,1//2,20,216,650,8.098328e6,3.013888888888889,0.020582507,8.497504e6,684,14,582,0,0
+24,5//1,1//2,30,582,1869,7.6582035353e10,3.2130584192439864,0.171186132,7.1555952e7,2740,17,2413,0,0
+25,6//1,1//2,42,1344,4497,4.013602463789026e15,3.3467261904761907,1.252807822,5.14580608e8,8574,20,7733,0,0
+26,7//1,1//2,56,2766,9520,1.0808789143539522e21,3.442154736080983,8.231138148,3.229056608e9,22498,23,20635,0,0
+27,8//1,1//2,72,5211,18308,1.4246720832694277e27,3.513529073114565,46.922879728,1.693481232e10,51816,26,48113,0,0
+28,9//1,1//2,90,9157,32679,8.887943639244825e33,3.568854428306214,226.27318923,7.5223131184e10,107982,29,101204,0,0
+29,10//1,1//2,110,15213,54963,-7.631638640409828e37,3.612962597778216,985.976136546,3.1699412416e11,207950,32,196322,0,0
+30,11//1,1//2,132,24135,88066,1.453474986156403e37,3.6489330847317176,3726.889421884,1.07794863488e12,375716,35,356784,0,0
+31,12//1,1//2,156,36842,135534,9.023451080536431e37,3.678817653764725,12588.248204712,3.191343287296e12,644052,38,614528,0,0
+32,1//1,1//4,3,4,5,3.0,1.5,0.000176371,20112.0,3,7,3,0,0
+33,2//1,1//4,12,108,426,48218.0,3.9537037037037037,0.003946962,1.901664e6,163,11,128,0,0
+34,3//1,1//4,31,4828,30377,1.934147339911883e16,6.292046396023198,6.211790538,2.913187312e9,24455,15,23192,0,0
+35,1//2,1//2,6,13,27,28.0,2.1538461538461537,0.000251611,76176.0,15,6,12,0,0
+36,1//2,1//4,12,93,374,37674.0,4.032258064516129,0.003455267,1.660064e6,139,9,109,0,0
+37,1//4,1//4,31,4755,34920,1.591733963543244e17,7.344058885383807,7.389094209,3.303960288e9,19020,10,17836,0,0
+38,4//1,2//1,42,1905,5980,6.742040747418136e15,3.1396325459317587,2.922999648,1.249701968e9,16108,30,14904,0,0
+39,0//1,2//1,6,16,24,20.0,1.5625,0.000416443,125856.0,29,11,24,0,0

examples/multiplication_tables.jl

@@ -24,8 +24,8 @@ ns1 = convert.(SurrealFinite, [0; 1:n2;        2;      1:n3;       1:n4;
 ns2 = convert.(SurrealFinite, [2; 2*ones(n2); -2; 3*ones(n3); 4*ones(n4); 1//2*ones(n_12); 1//4*ones(n_14); 1//2; 1//4; 1//4; 2; 2]) 
 # ns1 = convert.(SurrealFinite, [0:n2-1;           1:n3;       1:n4;         1:n_12;           1:n_14; 1//2; 1//2])
 # ns2 = convert.(SurrealFinite, [2*ones(n2); 3*ones(n3); 4*ones(n4); 1//2*ones(n_12); 1//4*ones(n_14); 1//2; 1//4]) 
-# ns1 = convert.(SurrealFinite, [0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 3; a]) 
-# ns2 = convert.(SurrealFinite, [2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 3; 2])
+# ns1 = convert.(SurrealFinite, [0; 1; 2; 3; 4; 5; 6; 7; 8; 3; a]) 
+# ns2 = convert.(SurrealFinite, [2; 2; 2; 2; 2; 2; 2; 2; 2; 3; 2])
 # ns1 = [2; 4]
 # ns2 = [1/2; 1/4] 
 n = length(ns1) 

src/SurrealFinite.jl

@@ -116,15 +116,15 @@ function convert(::Type{SurrealFinite}, n::Int )
     else
         result = SurrealFinite(string(n), ϕ, [convert(SurrealFinite, n+1)] )
     end
-    h = hash(result)
-    ExistingSurreals[h] = result
-    ExistingCanonicals[Rational(n)] = h
+    hr = hash(result)
+    ExistingSurreals[hr] = result
+    ExistingCanonicals[Rational(n)] = hr
     return result 
     # should check to make sure abs(n) is not too big, i.e., causes too much recursion
 end 
 function convert(::Type{SurrealFinite}, r::Rational )  
     global ExistingSurreals 
-    global ExistingCanonicals
+    global ExistingCanonicals 
     if haskey(ExistingCanonicals, r)  
         return ExistingSurreals[ ExistingCanonicals[r] ]
     elseif isinteger(r)  
@@ -143,7 +143,7 @@ function convert(::Type{SurrealFinite}, r::Rational )
     end 
     h = hash(result)
     ExistingSurreals[h] = result
-    ExistingCanonicals[r] = h
+    ExistingCanonicals[r] = h 
     return result
 end
 function convert(::Type{SurrealFinite}, f::Float64 ) 
@@ -249,8 +249,10 @@ promote_rule(::Type{T}, ::Type{SurrealFinite}) where {T<:Real} = SurrealFinite
 
 @static if VERSION < v"0.7.0"
     const ϕ = Array{SurrealFinite,1}(0) # empty array of SurrealFinites
+    const ∅ = ϕ 
 else
     const ϕ = Array{SurrealFinite,1}(undef,0) # empty array of SurrealFinites
+    const ∅ = ϕ
 end 
 const SurrealZero = SurrealFinite("0", ϕ, ϕ ) 
 const SurrealOne  = SurrealFinite("1", [ zero(SurrealFinite) ], ϕ ) 
@@ -388,11 +390,16 @@ function -(x::SurrealFinite)
         result = SurrealFinite(x.shorthand[2:end], -x.R, -x.L )
     else
         result = SurrealFinite("-"*x.shorthand, -x.R, -x.L )
-    end 
+    end
+    hr = hash(result)
+    if haskey(ExistingSurreals, hr)
+        result = ExistingSurreals[hr] # don't double up on memory
+    else
+        ExistingSurreals[hr] = result
+    end
     ExistingSurreals[hx] = x
-    ExistingSurreals[hash(result)] = result
     ExistingNegations[hx] = hash(result)
-    ExistingNegations[hash(result)] = hx
+    ExistingNegations[hr] = hx
     Count['-'] += 1
     return result
 end
@@ -440,7 +447,11 @@ function +(x::SurrealFinite, y::SurrealFinite)
         ExistingSums[hy] = Dict{UInt64,UInt64}() 
     end
     hr = hash(result)
-    ExistingSurreals[hr] = result
+    if haskey(ExistingSurreals, hr)
+        result = ExistingSurreals[hr] # don't double up on memory
+    else
+        ExistingSurreals[hr] = result
+    end    
     ExistingSums[hx][hy] = hr
     ExistingSums[hy][hx] = hr
     Count['+'] += 1
@@ -494,7 +505,11 @@ function *(x::SurrealFinite, y::SurrealFinite)
         ExistingProducts[hy] = Dict{UInt64,UInt64}()
     end 
     hr = hash(result)
-    ExistingSurreals[hr] = result
+    if haskey(ExistingSurreals, hr)
+        result = ExistingSurreals[hr] # don't double up on memory
+    else
+        ExistingSurreals[hr] = result
+    end
     ExistingProducts[hx][hy] = hr
     ExistingProducts[hy][hx] = hr
     Count['*'] += 1 
@@ -550,8 +565,8 @@ function expand(x::SurrealFinite; level=0)
     if level==0
         s = x.shorthand != "" ? x.shorthand : expand(x; level=1)
     elseif level==1
-        tmpL = isempty(x.L) ? "ϕ" : join(convert.(String, x.L), ',')
-        tmpR = isempty(x.R) ? "ϕ" : join(convert.(String, x.R), ',')
+        tmpL = isempty(x.L) ? "∅" : join(convert.(String, x.L), ',')
+        tmpR = isempty(x.R) ? "∅" : join(convert.(String, x.R), ',')
         s = "{ " * tmpL * " | " * tmpR  * " }"
         return s 
     elseif level>=1
@@ -567,10 +582,10 @@ function convert(::Type{SurrealFinite}, s::AbstractString )
     # interpret, (i) numbers as canonical, (ii) phi, \phi, ϕ correctly, (iii) structure
     @static if VERSION < v"0.7.0"
         s = replace(s, r"\s+", "") # remove white space
-        s = replace(s, r"phi|\\phi|ϕ|\{\}", "") # replace phi or \phi with empty set
+        s = replace(s, r"phi|\\phi|ϕ|∅|\{\}", "") # replace phi or \phi with empty set
     else
         s = replace(s, r"\s+" => "") # remove white space
-        s = replace(s, r"phi|\\phi|ϕ|\{\}" => "") # replace phi or \phi with empty set
+        s = replace(s, r"phi|\\phi|ϕ|∅|\{\}" => "") # replace phi or \phi with empty set
     end
     # println("     s1 = $s")
     not_end = true
@@ -627,13 +642,15 @@ function surreal2tex(io::IO, x::SurrealFinite; level=0)
     s = expand(x; level=level)
     @static if VERSION < v"0.7.0"
         s = replace(s, r"\{", " \\{ ") 
-        s = replace(s, r"ϕ", " \\phi ") 
+        s = replace(s, r"ϕ", " \\emptyset ") 
+        s = replace(s, r"∅", " \\emptyset ") 
         s = replace(s, r"\|", " \\mid ") 
         s = replace(s, r"\}", " \\} ")
         s = replace(s, r"(\d+)//(\d+)", s"\\frac{\1}{\2}")
     else
         s = replace(s, r"\{" => " \\{ ") 
-        s = replace(s, r"ϕ" => " \\phi ") 
+        s = replace(s, r"ϕ" => " \\emptyset ") 
+        s = replace(s, r"∅" => " \\emptyset ") 
         s = replace(s, r"\|" => " \\mid ") 
         s = replace(s, r"\}" => " \\} ")
         s = replace(s, r"(\d+)//(\d+)" => s"\\frac{\1}{\2}")
@@ -660,7 +677,7 @@ end
 # show(io::IO, X::Array{SurrealFinite}) = print(io, "{", join(X, ", "), "}")
 function show(io::IO, X::Array{SurrealFinite})
     if isempty(X)
-        print(io, "ϕ")
+        print(io, "∅")
     else
         print(io, join(X, ", "))
     end

src/SurrealNumbers.jl

@@ -16,15 +16,15 @@ export clearcache
 # export SurrealAlt2  
 # export ExistingSurrealsAlt2, ExistingProductsAlt2, ExistingSumsAlt2
 
-export <=, zero, one, ==, ≅, ≇, <, -, +, *, ≀, ↑, ↓, dag_add,
+export ∅, ϕ, <=, zero, one, ==, ≅, ≇, <, -, +, *, ≀, ↑, ↓, dag_add,
     dali, convert, promote, hash, 
     pf, pff, spf, show, surreal2node, surreal2dot, surreal2dag, surreal2tex, read, expand,
     generation, canonicalise, iscanonical, parents, isancestor, ≺, ≻, ⪯, ⪰, 
     unique2!, 
     sign, round, floor, ceil, trunc, mod, 
     isinteger, isinf, isnan, isfinite, isodd, iseven, isdivisible,
     dag_stats, nodes, edges, paths, tree_nodes, breadth, width, surrealDAG
-
+  
 abstract type Surreal <: Real end 
 
 @static if VERSION < v"0.7.0"

src/todo.txt

@@ -1,9 +1,6 @@
 
-*** replace caches with dict -> another hash, and have a global list
-    of all surreals to
-      (a) reduce memory footprint
-      (b) avoid the problem that 1+3 == 2 +2, but at the moment might
-          point to new memory
+- use a hash based cache in DAG stats, so that we avoid doubling up if
+  by mistake the same surreal is allocated twice
 
 - put "values" into surreals in the same way as hashes,
      - but for surreals resulting from computations this should be easy 

test/Data/test_surreals_io.tex

@@ -1,3 +1,3 @@
 \frac{13}{16}
  \{  3/4  \mid  7/8  \} 
- \{   \{   \{   \{   \phi   \mid   \phi   \}   \mid   \{   \{   \phi   \mid   \phi   \}   \mid   \phi   \}   \}   \mid   \{   \{   \phi   \mid   \phi   \}   \mid   \phi   \}   \}   \mid   \{   \{   \{   \{   \phi   \mid   \phi   \}   \mid   \{   \{   \phi   \mid   \phi   \}   \mid   \phi   \}   \}   \mid   \{   \{   \phi   \mid   \phi   \}   \mid   \phi   \}   \}   \mid   \{   \{   \phi   \mid   \phi   \}   \mid   \phi   \}   \}   \} 
+ \{   \{   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \emptyset   \}   \}   \mid   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \emptyset   \}   \}   \mid   \{   \{   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \emptyset   \}   \}   \mid   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \emptyset   \}   \}   \mid   \{   \{   \emptyset   \mid   \emptyset   \}   \mid   \emptyset   \}   \}   \}