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A Simple Algorithm And Proof For Type Inference

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ž < M @ 9 Z Z ž < 9 6  : < [ S > < 9 Z , <  B > 8 < Á @ Z  4 ; 6 M @  ´ 7 = n < J = 4  4 Y 6 7 4  B ß K , 8 Ê = @ 9 Z ¼ < J 4 h B C @ J = Ë = 7 4  B D w 8 Y 6  4 2 ^ ¼ <  D < ¹ 4JB]476Y4 H BZYÕw½  R O C H L O H  ?  < M @ [ Z Z <  < « : · @ 2 ^ ß < Á @ 9 Z w < 6 [ Z _ ´ 7 =  Z  Ó · 8 C @ J = J = , < D 9 z Õ  i M @  B 5 D e @ [ Z , < 6 µ HML žÓˆ6`_ P 4J8>@M=J=_ §Õw½ -  # Å ±²:]<Z[<,6 HML †4JZ²DD54JBD R 8,6947^2<,=7_»´/_ ӌjÕ Ól Õ Ó >Õ M H L H Q O H L      #   ² ± ] : ® < 9 Z w < 6 ’ J 4 Z ] ´ ; 6 C @ J 4  B , < + D f ´ _  D 4 0 _ J 4  B K Å 6Y:]< OQÓ H D]4 L ³ _04JBåK»Õ 8>=Ê@ R Z9<½ Ó a Õn47BÒ69:@MD ³                  "! &' () *! 0 KJMLN'JOP-Q-Q- :,<4>@DC63FE?7=3G1HI@45   BA #$ % ,+ .-/    21436587#9;:=@45 BA& :=@45 %0RSA UTLN'TVO WAX!   AX! #+g \] T?-/   ] YTL%Z[TVO %(A :=<4>@'^_H`3Fa14^G3Fbc:c@8E?ad5   e ] ' eX 'Tf M H  L L C H L O H R R ± ²  :  < 9 Z , < 6 9 S , < ž < C ^ M @ [ S Ê 4 M @ ] ´ J = > < Z Á @ Y 6 , < S 7 4 B S 7 4 ’ Z  D , < « ¾ ] B > <  D 7 4  B Y 6  : < Z M @ = · @ M _ ½ µ W H 6Y<®6Y:@C6¼6Y:BBf69DZ <,D»HCL 6 H <><>Z9DÒZ¥B6Y:H <6žP ´#S <ÞHMPFÐ µH Z[<>47=J69= 4¹.H 6`_BÁP 6Y<>:D@CÑ6†47@¼B¬6Y<,69:S O < R 4JZ Z R @M<>= =7=µ ¹.6`@_ _MP ½e<>Dg ³ µ P L R H ³      #   g J 4 N Z @ M @ [ S 9 6 Ê 4 M @ =  B w 8 Y 6 4 B  4 Y 6  :  ¾  B 7 4 9 6 < Å µ D H2O @M47B½ P P      #   g 7 4 h Z A @ Y 6 9 S 4 J = È < Ó [ w Õ i  : > < [ S < Ç 7 4 h Z È @ ` 6 _ < Å µ @6YM4<¼6Y:D <,H2S¸O 6`_@C4JP B <>D HCL H åS R 47Z†Bf6`47Z¹ µ HML ½ H H H H O ² ±  : h < Ê = M @ X Z ® 6 8  B  D  4 Y 6 4 A B J 4 A B 9 6  : J 4 ® Z ] D , < « ¾  B  4 Y 6 4 È B J 4 Á Z  @ 9 6 > < ; 8  : ] B J 4 · 8 C @ =  B <  : 7 4 ; 8 : M @ C à > < Á Z Y 6 ] : < µ P S HVHML Z²Z[=J47K2:f6Y=7_ÀB<>@C6Y<,S>½ L H H L C H L Ü  ? < 9 S  4 Y 6 < S 9 S , < [ Z 9 6 9 S J 4 w 8 Y 6 , <  D 6 Y 6  : < 9 S , < ¸ < C ^ M @ [ S Ê 4 M @ ] ´ J = > < Z ½ Þ ? < 9 S 7 4 9 6 < Á Ö µ µ L 6YH :]S²<ž6Y6`:]_ <P < BZ[4 H B HML 6Y:]< L R B]8,6Y4 H B 47B µ :478;:Â6Y:]<^C@MS[4Ê@M´=7< Ì4JZU@MZ[Z H 8>4Ê@6Y<>D µ 4769: P R L H P O R R L  ? ¼ < ] B w < 0 à ! 6 2 K 7 4 M ^ n < Y 6  : n < ` 6 _ 7 4  B e K S 7 = > < Z I S 9 6  : n < ` 6 _ , < 5 D J = @ # ´ « D ® @ · 8 C @ J = 8 = , Z ½ 9 6  : ž < M @ [ Z 9 Z > < Œ S ¹ 6Y6Y4:H @CBÏ6 Ó :«@CZ 6`[_ÕnP 4J´S)=7<†Z>Z9i :µ @M< =J=0µZ9<,S9S O 47BZ HML 6Y:< L S9<,<¼^M@CS94Ê@C´=J<,Z²4JB µ ½ ±²:<ž6`_ P 4JBKeS R =J<>Z L H =7= H µ 6Y:]<D]<,¾«B46Y4 H B HML ³ ³ µ :<,S9< ÁÓ FÕ ³ ³ ÞÓ +Õ ³ ³ Ó ÁÖ Ü Õ AÓ ³ Õ ³ ±6Y²::@C<Á6’6`476’_ P 47B/Z!OQ@eP 69=J4<>H K2BÌ@M=F47B+6`_ 6YP :<Z[½ 4JZ HML 6Y:<Á=J@MZ[6nS R =7<Á4JZnD<,Z947K2B<>D¬6 H <,BZ R S9< H SÁL 6`H _ P <+47B L <>S[<>B8,<,S9B<,D P µ 4769: 6Y:]<ÂH 8 H S9L S9H <,Z P#R H BDP47BKD]<>D O R 8w6Y4 H B Ø O Z[_0:ZX476Y8;< : 69:<¼S 6`_ 'P i†<474JB B L µ H S:O 478;@: 6Y4 µ H B»<5:«D@M<·Z’@C= ´#<>µ <,4B56Y: <,SY6`@M_ Z[<><ÌD ³ @MZ[Z9<>SX6Y4 BZ S Bf6`_ <>D6Y<,S Z>i²47B  ]   ] :=@\36545N@8E?:,7      :=@&36545 a&> =: 7      :=@  Z[A d4'T 36545N@8E?:=7     T        T \T  d4'T S  T  T   ]   Y T T  W µ  TVOPW TL Z  TL   TL%!  TVO ! #+g  #$ TLQ-  TVO TL Z[TVO  ³ :<>S[< ®Ó FÕ µ ³ ³ AÓ ÂÕ ³ ³ Ó ÁÖ Ü Õ ÞÓ åÕ ³ P L P H O L H H O O L H =8·H @Cµ B5Z ±²8;³ž:«:É @C<SY4@M^26`_8w<,6YB¬<><S[@h4 4J,B 8,<ž= <>H 69S9Z[:<,<><>BDÌZ[8><<6Y<,6`_S O PS <>´Z =J< B= O DZ9<,<>@CDÂ8;=J:«=u6Y@C6`<,B_ S 4JP Z <>O Z;Z Õ HMUL Z @·R _8;:¬6Y:6Y<,:B+@C6 ´#<Z[6Y@C6Y<,³D̳ @Mz½ßZ ?Þ=< ¹ HML R -  # @MDD]4JBKÁ6 H 69:<žDÅ<,¾«B]±²476Y:4 <ÌH B Z9DÙ4JBD 8,6Y4^2<,=7_ ´/_ ò _    Y XT T    TVOP  TLZ TL (  W TVOB!  TL !   #+g .-  TL Z \T $#   #  TVO TVO !"  :,<4>@ @&% > E@454587  5 _A' A :,<4>@ @&% > E@454587  5 \A& SA(' T H H P P H H  P P P H ç ?  < 7 4 J = ) = M @ 7 = Z M @ 7 = = ` 6 _ À < z < à 9 S , < 9 Z 9 Z 4  B ž Z 6 @ · < M @ ž S 7 4 ¬ B ` 6 _ » < C @ 9 Z 9 Z , < [ S 9 6 4  B Z À < > 8 M @ J = = µ µ µ 6Y:]<S[<>Z R =7694JBK H ´[<,8,69Z w½ RL H ´ZX6Y476 R 6Y4 H H BZHM L L S H2O R 6`_ P H Z9Z[4 H B]Z¥P 6 H 6`_ PP <,ZI4JBeH 6Y:]< R Z R @M= µ L @·H _Mi  ? ! <  D w < « ¾  B ' < Z @47M69BZ’D @M8,µ 69S[4 H476YB < HB69:SI<¼6Y6`:_ <žP <ß@C8,@M6YZ94 Z RB OÀP 6Y@Á4 H Z B ´ßZ[69i47K26 4^26Y<>4 B B ´f_ Bh@®Ú6`Ó _ FÕ <¼ºFZ[Õ[Z9Õ4 NB½ i@MBD  S ?< O @·_%B H µ ZX6;@C69<ž69:< O @C4JB»69:< H S[< O ³ $u /# 0 Å        ³          D:=<4>@8b'@&% > E4@454587  36545N@8E8:=7  5  T     Y 7 CF@  3_98H  5N@'^K:c@8E8a 7   7 :7I5D^ @4987=^G3G1HI@ @8: @8E\: @8E@ &@ % 7I58: 5 3(:,<4>@DTf5 9 2: 3F: ! S T : @8E@7I5 3  < 5 9 _T &: @  : @8E@7I5 3(:=@ @&% > E@454587 ! 59): 3F:&:@:,<4> 7 " 5    3FE@ > E@4987I5N@8H <_:@:=@45   :@    ED3FH H 5 1858:,7 : :=7  5#   E?a R M H L R R  R P H 2 H Q O H T ; 8 : @ J 4 ¥ Z · 8 M @ 7 = J = > < % D @ ½ .6 † J 4 I Z , 8 J = > < M @ 9 S  = _  B 4 < 6 J 4 Z X S ¹ P :<>47=JZ=Ÿ½O ½'?Þ< µ 4J=J=Ÿi HML 8 H2R S[Z9Zn@CZ µ $Å $u Ì   &% 0  f# HMR L @M=J±²= :PFZ9<,47B/´69=J<¼SYP @MD= <,P S94S ^MH H @´6Y=74 < H O BZ>4J½)BÒ±²@M:Bf47_ÌHMZ’L 4J@MZ’=7KD P H H S[B47<ž6Y: H´fO _hL Z9H 4 O®P SnR6`_ =J@CP 6YK26 S9H 4^CàM@C<,6Y< P 4 PH B69iSY´@M8;R à 6 ZO [4J4JK2B:fK¼6 6`H_ 8>8 <·Z[@MZ98;4 :5BB ZIH 47DB< Z[69HM<·L@MD 69:<žD6`_ <>S[47<>^CZÚ@C696 4 H B»àC<>69< S9<,ZÚ6Y:«@6 ±P ²:<ž@M=7K H HMS9L 46Y: O O 4 O HM4JL 8,Z¸R 69:<¼H 8 H BZ[69S R 8w6Y4 H B HML 6Y:P <¼D<,S9P 4^M@6Y4 H H B½ g†6’HSX_Z[69< H P 476 àMP <,S Z!D6Yi2S9@M@M8;Bà D®Z[<,6 @eZ9i <,µ <>:B®478;6`:+_ P @M<,i Z9Z[4:47BÌ8;:®@C@MZ9Z9S[<,<¥S[^M694<>S9B4¾«Z’8·6@6Y4 H´#B < 8K2H4^2B<¼D476Y69:4 <žH B]ZXZ 6YS µ R :8,6 478;R : S9< O®HMR L ZX6Y6I:´#<<'@MZ9=7K@CH6Y47S9Z[4¾«6Y: <,DO µ L ³ H SÚ6Y:S[47^C@C694 H Be6 H ´Fµ K2@M="½u?< O @·_eB H µ  #ˆ- )z* % V f#, + -Úg¶6Y<>S O /. 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