ordinal_games.log

This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Arch Linux) (preloaded format=pdflatex 2016.8.21)  11 OCT 2016 16:07
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**ordinal_games
(./ordinal_games.tex
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\OT1/cmr/m/n/10.95 Malacaria [[]], by Hy-land and Ong [[]], and by Nickau [[]])
. 
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[]
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[] \OT1/cmr/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 s \OMS/cmsy/m/
n/10.95 2 \OML/cmm/m/it/10.95 M[]  [] []  []\OMS/cmsy/m/n/10.95 g
 []


Overfull \hbox (12.65497pt too wide) in paragraph at lines 356--357
[]\OT1/cmr/m/n/10.95 It fol-lows im-me-di-ately from the def-i-ni-tions that $[
][][]\OML/cmm/m/it/10.95 A$ \OT1/cmr/m/n/10.95 is a well formed game. 
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[]\OT1/cmr/m/n/10.95 (ii): $[] []$ 
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Overfull \hbox (0.40103pt too wide) in paragraph at lines 493--495
\OT1/cmr/m/n/10.95 if $\OML/cmm/m/it/10.95 A; B$ \OT1/cmr/m/n/10.95 are neg-a-t
ive games. In other words, we can ig-nore the well-formedness
 []

[19]
Overfull \hbox (3.58801pt too wide) detected at line 527
\OMS/cmsy/m/n/10.95 f\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 s\OMS/cmsy/m/n/10.
95 j[]\OML/cmm/m/it/10.95 ; s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 )  [] []
  []\OMS/cmsy/m/n/10.95 g \OT1/cmr/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OT1/cmr/m/
n/10.95 (\OML/cmm/m/it/10.95 s[]\OMS/cmsy/m/n/10.95 j[]\OML/cmm/m/it/10.95 ; s[
]\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 )  [] []  []\OMS/cmsy/m/n/10.95 g
 []

[20]
Overfull \hbox (2.31253pt too wide) detected at line 565
[]\OT1/cmr/m/n/10.95 (\U/euf/m/n/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.
95 ) \OMS/cmsy/m/n/10.95 ^^T []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 s\OT1/cm
r/m/n/10.95 ) = \OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 ^^@ []\OT1/cmr/m/n/10
.95 (\OML/cmm/m/it/10.95 t\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 ) \OMS/cmsy
/m/n/10.95 ^^T \OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 ^^@ []\OT1/cmr/m/n/10.
95 (\U/euf/m/n/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 ) = [](\U/euf/m
/n/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 )
 []

[21] [22] [23] [24] [25]
Overfull \hbox (7.85733pt too wide) in paragraph at lines 715--716
\OT1/cmr/m/n/10.95 win-game, then a \OT1/cmr/m/it/10.95 win-ning strat-egy \OT1
/cmr/m/n/10.95 for $\OML/cmm/m/it/10.95 A$ \OT1/cmr/m/n/10.95 is a to-tal strat
-egy for $(\OML/cmm/m/it/10.95 M[]; ^^U[]; ^^P[]; P[]\OT1/cmr/m/n/10.95 )$
 []

[26] [27] [28] [29] [30] [31]
Overfull \hbox (22.26468pt too wide) in paragraph at lines 894--895
[]\OT1/cmr/m/n/10.95 Setting $\OML/cmm/m/it/10.95 M[] \OT1/cmr/m/n/10.95 = \OML
/cmm/m/it/10.95 M[] \OT1/cmr/m/n/10.95 = \OML/cmm/m/it/10.95 M[] \OT1/cmr/m/n/1
0.95 = \OML/cmm/m/it/10.95 M[] \OMS/cmsy/m/n/10.95 t \OML/cmm/m/it/10.95 M[]$\O
T1/cmr/m/n/10.95 , and re-strict-ing $\OML/cmm/m/it/10.95 ^^P[]; ^^P[]; ^^P[]$
 []


Overfull \hbox (11.05331pt too wide) in paragraph at lines 903--905
\OT1/cmr/m/it/10.95 over, $\OML/cmm/m/it/10.95 P[]$ \OT1/cmr/m/it/10.95 is the 
largest sub-set $\OML/cmm/m/it/10.95 X \OMS/cmsy/m/n/10.95 ^^R \OML/cmm/m/it/10
.95 P[]\OMS/cmsy/m/n/10.95 k\OML/cmm/m/it/10.95 P[]$ \OT1/cmr/m/it/10.95 such t
hat $\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 M[]; ^^U[]; ^^P[]; X\OT1/cmr/m/n/1
0.95 )$
 []

[32] [33] [34] [35]
Overfull \hbox (5.74709pt too wide) in paragraph at lines 982--983
\OML/cmm/m/it/10.95 s\OMS/cmsy/m/n/10.95 j[]$ \OT1/cmr/m/n/10.95 and $\OML/cmm/
m/it/10.95 s\OMS/cmsy/m/n/10.95 j[]$ \OT1/cmr/m/n/10.95 are both lim-it-ing pla
ys. By Lemma 3.6[], $[](\OML/cmm/m/it/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/
n/10.95 )\OML/cmm/m/it/10.95 ; []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 s\OMS/
cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^T
 []

[36] [37] [38] [39] [40]
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[]|$[]$|  
 []


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[]|$[]$|  
 []


Overfull \hbox (6.9344pt too wide) detected at line 1129
[]
 []

[41] [42] [43] [44] [45] [46] [47] [48]
Overfull \hbox (35.44717pt too wide) in paragraph at lines 1361--1361
[] 
 []


Overfull \hbox (86.78394pt too wide) in paragraph at lines 1393--1393
[] 
 []

[49] [50] [51] [52]
Overfull \hbox (1.36723pt too wide) in paragraph at lines 1538--1540
\OMS/cmsy/m/n/10.95 F [] [] G\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 \OT1/cmr/
m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! []$\OT1/cmr/m/n/10.95 . The re-la-tional con-
tent $[]$ of the strat-egy $[][]$
 []


Overfull \hbox (40.22849pt too wide) in paragraph at lines 1548--1548
[] 
 []

[53]
Overfull \hbox (20.00883pt too wide) in paragraph at lines 1560--1561
[]\OT1/cmr/m/n/10.95 We there-fore have the struc-ture of a sym-met-ric monoida
l cat-e-gory on $\OMS/cmsy/m/n/10.95 G\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 
\OT1/cmr/m/n/10.95 )$. 
 []

[54] [55] [56] [57] [58] [59] [60] [61] [62]
Overfull \hbox (16.99615pt too wide) in paragraph at lines 1820--1821
\OT1/cmr/m/n/10.95 both ap-pro-pri-ate sub-sets of $\OML/cmm/m/it/10.95 P[]\OMS
/cmsy/m/n/10.95 k\OML/cmm/m/it/10.95 P[]\OMS/cmsy/m/n/10.95 k\OML/cmm/m/it/10.9
5 P[]$\OT1/cmr/m/n/10.95 . We just need to check that $\OML/cmm/m/it/10.95 ^^U[
] \OT1/cmr/m/n/10.95 =
 []


Overfull \hbox (10.48459pt too wide) in paragraph at lines 1838--1839
[]\OT1/cmr/m/n/10.95 To con-struct the monoidal closed struc-ture, it suf-fices
 to give a mor-phism 
 []

[63]
Overfull \hbox (39.49551pt too wide) detected at line 1884
[]\OMS/cmsy/m/n/10.95 k [][] \OT1/cmr/m/n/10.95 = []
 []

[64]
Overfull \hbox (74.48236pt too wide) detected at line 1899
[] \OMS/cmsy/m/n/10.95 k [][] \OT1/cmr/m/n/10.95 = []
 []

[65] [66]
Overfull \hbox (0.44365pt too wide) in paragraph at lines 1987--1988
[]\OT1/cmr/m/n/10.95 Now by al-ter-na-tion we must have ei-ther $\OML/cmm/m/it/
10.95 ^^P[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1
/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 a; m\OT1/cmr/m/n/10.95 )) = \OML/cmm/m/it/1
0.95 O$ \OT1/cmr/m/n/10.95 or $\OML/cmm/m/it/10.95 ^^P[]\OT1/cmr/m/n/10.95 (\OM
L/cmm/m/it/10.95 s\OMS/cmsy/m/n/10.95 j[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.9
5 b; n\OT1/cmr/m/n/10.95 )) =
 []

[67] [68] [69] [70]
Overfull \hbox (77.01457pt too wide) in paragraph at lines 2108--2108
[] 
 []

[71] [72]
Overfull \hbox (0.96515pt too wide) in paragraph at lines 2152--2153
\OT1/cmr/m/n/10.95 our se-quoidal cat-e-gory is \OT1/cmr/m/it/10.95 de-com-pos-
able \OT1/cmr/m/n/10.95 and the iso-mor-phism $[][] [] [] (\OML/cmm/m/it/10.95 
A \OMS/cmsy/m/n/10.95 ^^B
 []

[73] [74]
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[] 
 []


Overfull \hbox (51.26396pt too wide) detected at line 2260
\OML/cmm/m/it/10.95 P[] \OMS/cmsy/m/n/10.95 \ []
 []


Overfull \hbox (30.56708pt too wide) detected at line 2268
\OML/cmm/m/it/10.95 S \OT1/cmr/m/n/10.95 = \OML/cmm/m/it/10.95 P[] \OMS/cmsy/m/
n/10.95 \ []
 []

[75]
Overfull \hbox (10.21413pt too wide) detected at line 2298
\OT1/cmr/m/n/10.95 ([][] \OMS/cmsy/m/n/10.95 \OML/cmm/m/it/10.95 ^^\\OT1/cmr/m
/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^N \OML/cmm/m/it/10.95 ^^[ \OT1/cmr/m/n/10.95 =
 []
 []

[76]
Overfull \hbox (1.35188pt too wide) in paragraph at lines 2325--2327
\OT1/cmr/m/n/10.95 By con-struc-tion, the $\OML/cmm/m/it/10.95 A$\OT1/cmr/m/n/1
0.95 -components of the $\U/euf/m/n/10.95 t[]$ \OT1/cmr/m/n/10.95 agree (in-dee
d, the $\OML/cmm/m/it/10.95 A[]$\OT1/cmr/m/n/10.95 -component
 []

[77] [78]
Overfull \hbox (62.06122pt too wide) detected at line 2409
[]
 []


Overfull \hbox (17.37387pt too wide) detected at line 2414
\OML/cmm/m/it/10.95 B [] B \OMS/cmsy/m/n/10.95 
 \OML/cmm/m/it/10.95 B [] A \OMS/cmsy/m/n/10.95 
 \OML/cmm/m/it/10.95 B [] A \OMS/cmsy/m/n/10.95  \OML/cmm/m/it/10.95 B [] \OT1
/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 A \OMS/cmsy/m/n/10.95  \OML/cmm/m/it/10.95
 B\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^B \OT1/cmr/m/n/10.95 (\OML/cmm/m/i
t/10.95 A \OMS/cmsy/m/n/10.95  \OML/cmm/m/it/10.95 B\OT1/cmr/m/n/10.95 ) \OMS/
cmsy/m/n/10.95 ! []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 A \OMS/cmsy/m/n/10.9
5 ^^B \OML/cmm/m/it/10.95 A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95  \OML/cmm/
m/it/10.95 B
 []

[79] [80] [81] [82] [83]
Overfull \hbox (8.06529pt too wide) in paragraph at lines 2641--2641
[]\OT1/cmr/bx/n/12 The rank of a trans-fi-nite se-quence of nat-u-ral num-bers 

 []

[84] [85] [86] [87] [88] [89] [90] [91]
Overfull \hbox (1.96648pt too wide) in paragraph at lines 2859--2861
\OT1/cmr/m/n/10.95 As we have seen, this set is prefix-closed, so $(!\OML/cmm/m
/it/10.95 A\OT1/cmr/m/n/10.95 )[]$ is a well formed game. 
 []

[92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105]
[106] [107] [108] [109] [110] [111]
Overfull \hbox (17.5971pt too wide) in paragraph at lines 3419--3420
\OML/cmm/m/it/10.95 ^^\[] [] [] []B \OMS/cmsy/m/n/10.95 ! []\OML/cmm/m/it/10.95
 C$ \OT1/cmr/m/n/10.95 then we get a weak-en-ing $\OML/cmm/m/it/10.95 B[]$ \OT1
/cmr/m/n/10.95 of $\OML/cmm/m/it/10.95 B$ \OT1/cmr/m/n/10.95 and an iso-mor-phi
sm $(\OML/cmm/m/it/10.95 ^^\[]\OT1/cmr/m/n/10.95 )[] [] [] []\OML/cmm/m/it/10.9
5 B[] \OMS/cmsy/m/n/10.95 !
 []

[112] (./ordinal_games.bbl [113]
Underfull \hbox (badness 1127) in paragraph at lines 9--12
\OT1/cmr/m/n/10.95 Full ab-strac-tion for PCF.  \OT1/cmr/m/it/10.95 In-for-ma-t
ion and Com-pu-ta-tion\OT1/cmr/m/n/10.95 ,
 []

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