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ind : Type.
1 : ind.
2 : ind.
3 : ind.
4 : ind.
5 : ind.
6 : ind.
7 : ind.
8 : ind.
9 : ind.
X : ind.

def P : ind -> ind.
[] P 1 --> X.
[] P 2 --> 1.
[] P 3 --> 2.
[] P 4 --> 3.
[] P 5 --> 4.
[] P 6 --> 5.
[] P 7 --> 6.
[] P 8 --> 7.
[] P 9 --> 8.

def S : ind -> ind.
[] S 1 --> 2.
[] S 2 --> 3.
[] S 3 --> 4.
[] S 4 --> 5.
[] S 5 --> 6.
[] S 6 --> 7.
[] S 7 --> 8.
[] S 8 --> 9.
[] S 9 --> X.

def trunc3 : ind -> ind.
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[]  trunc3 1 --> 1
[]  trunc3 2 --> 1
[]  trunc3 3 --> 1
[]  trunc3 4 --> 4
[]  trunc3 5 --> 4
[]  trunc3 6 --> 4
[]  trunc3 7 --> 7
[]  trunc3 8 --> 7
[]  trunc3 9 --> 7.
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bool : Type.
T : bool.
F : bool.

def and : bool -> bool -> bool.
[x] and T x --> x
[x] and F x --> F.

def or : bool -> bool -> bool.
[x] or T x --> T
[x] or F x --> x.

def not : bool -> bool.
[] not T --> F
[] not F --> T.

def eq : ind -> ind -> bool.
[x]   eq x x --> T
[x,y] eq x y --> F.




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line : Type.
l : ind -> ind -> ind -> ind -> ind -> ind -> ind -> ind -> ind -> line.
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def sudo : Type.
def c : line -> line -> line -> line -> line -> line -> line -> line -> line -> sudo.

def getl : ind -> line -> ind.
[x] getl 1 (l x _ _ _ _ _ _ _ _) --> x
[x] getl 2 (l _ x _ _ _ _ _ _ _) --> x
[x] getl 3 (l _ _ x _ _ _ _ _ _) --> x
[x] getl 4 (l _ _ _ x _ _ _ _ _) --> x
[x] getl 5 (l _ _ _ _ x _ _ _ _) --> x
[x] getl 6 (l _ _ _ _ _ x _ _ _) --> x
[x] getl 7 (l _ _ _ _ _ _ x _ _) --> x
[x] getl 8 (l _ _ _ _ _ _ _ x _) --> x
[x] getl 9 (l _ _ _ _ _ _ _ _ x) --> x.

def getc : ind -> sudo -> line.
[x] getc 1 (c x _ _ _ _ _ _ _ _) --> x
[x] getc 2 (c _ x _ _ _ _ _ _ _) --> x
[x] getc 3 (c _ _ x _ _ _ _ _ _) --> x
[x] getc 4 (c _ _ _ x _ _ _ _ _) --> x
[x] getc 5 (c _ _ _ _ x _ _ _ _) --> x
[x] getc 6 (c _ _ _ _ _ x _ _ _) --> x
[x] getc 7 (c _ _ _ _ _ _ x _ _) --> x
[x] getc 8 (c _ _ _ _ _ _ _ x _) --> x
[x] getc 9 (c _ _ _ _ _ _ _ _ x) --> x.




def setl : ind -> ind -> line -> line.
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 1 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l k  x2 x3 x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 2 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 k  x3 x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 3 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 k  x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 4 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 k  x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 5 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 x4 k  x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 6 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 x4 x5 k  x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 7 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 x4 x5 x6 k  x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 8 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 x4 x5 x6 x7 k  x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setl 9 k (l x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (l x1 x2 x3 x4 x5 x6 x7 x8 k ).

def setc : ind -> (line -> line) -> sudo -> sudo.
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 1 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c (k x1) x2 x3 x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 2 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 (k x2) x3 x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 3 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 (k x3) x4 x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 4 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 (k x4) x5 x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 5 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 x4 (k x5) x6 x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 6 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 x4 x5 (k x6) x7 x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 7 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 x4 x5 x6 (k x7) x8 x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 8 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 x4 x5 x6 x7 (k x8) x9)
[k,x1,x2,x3,x4,x5,x6,x7,x8,x9] setc 9 k (c x1 x2 x3 x4 x5 x6 x7 x8 x9) --> (c x1 x2 x3 x4 x5 x6 x7 x8 (k x9) ).
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def E : line := l X X X X X X X X X.
def empty_sudo : sudo := c E E E E E E E E E.

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def get : ind -> ind -> sudo -> ind.
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[i,j,s] get i j s --> getl j (getc i s).
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def set : ind -> ind -> ind -> sudo -> sudo.
[i,j,k,s] set i j k s --> setc i (setl j k) s.
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def exists' : (ind -> bool) -> ind -> bool.
[f  ] exists' f X --> F
[f,i] exists' f i --> or (f i) (exists' f (S i)).

def exists : (ind -> bool) -> bool := f => exists' f 1.


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def mem_line : ind -> ind -> ind -> sudo -> bool.
[i,j,k,s] mem_line i j k s --> exists (i' => eq (get i' j s) k).

def mem_col  : ind -> ind -> ind -> sudo -> bool.
[i,j,k,s] mem_col i j k s --> exists (j' => eq (get i j' s) k).

def mem_squ  : ind -> ind -> ind -> sudo -> bool.
[i,j,k,s] mem_squ i j k s -->
  (i':ind =>
   j':ind =>
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    (or (eq (get i'         j'         s) k)
    (or (eq (get (S i')     j'         s) k)
	(or (eq (get (S (S i')) j'         s) k)
    (or (eq (get i'         (S j')     s) k)
	(or (eq (get (S i')     (S j')     s) k)
	(or (eq (get (S (S i')) (S j')     s) k)
    (or (eq (get i'         (S (S j')) s) k)
	(or (eq (get (S i')     (S (S j')) s) k)
	    (eq (get (S (S i')) (S (S j')) s) k)))))))))
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  )
  (trunc3 i) (trunc3 j).


def check : ind -> ind -> ind -> sudo -> bool :=
i => j => k => s =>
     and (not (mem_line i j k s))
   ( and (not (mem_col  i j k s))
         (not (mem_squ  i j k s))).


solution : Type.
success : sudo -> solution.
fail : solution.


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def ite : bool -> solution -> solution -> solution.
[x] ite T x _ --> x
[x] ite F _ x --> x.

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def iffail : solution -> solution -> solution.
[x] iffail (success x) _ --> success x
[d] iffail fail     d --> d.

def solve : ind -> ind -> ind -> sudo -> solution.
[i,s]     solve i X _ s --> solve (S i) 1 1 s.
[  s]     solve X _ _ s --> success s.
[i,j,s]   solve i j X s --> fail.
[i,j,k,s] solve i j k s -->
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  ite
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    (eq (get i j s) X)
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    (ite
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      (check i j k s)
      (iffail
        (solve i (S j) 1 (set i j k s))
        (solve i j (S k) s)
      )
      (solve i j (S k) s)
    )
    (solve i (S j) 1 s).


def solve_sudo : sudo -> solution := s => solve 1 1 1 s.