var b : Type

var a : Type+

type List(a : Type+)

free type List(a : Type+) ::= Nil |

Cons (head : a; tail : List a)

free type List(a : Type+) ::= Nil |

Cons (head : a; tail : List b)

free type List(a : Type+) ::= Nil |

Cons (head : a; tail : List a b)

free type List(a : Type+) ::= Nil |

Cons (head : a; tail : List (List a))

free type Tree(a : Type+) ::= Leaf |

Branch (head : Tree a; tail : Tree a)

generated {type Tree1(a : Type+) ::= Leaf |

Branch (head : Tree a; tail : Tree1 a)}

type Tree2(a : Type+) ::= Leaf |

Branch (head : Tree a; tail : Tree2 a)

free type even(a : Type+) ::= rek (odd a);

odd(a : Type+) ::= rek (even a)

free type even2(a : Type+) ::= rek (odd2 (odd2 a));

odd2(a : Type+) ::= rek (even2 a)

%% Type Constructors -----------------------------------------------------

List

: Type+ -> Type

%[free type List(a : Type+)

::= Nil : List a_v-1

Cons : a_v-1 * List a_v-1 -> List a_v-1

(head : a_v-1; tail : List a_v-1)]%

Logical : Type := Unit ->? Unit

Pred : Type -> Type := \ a : Type . a_v-1 ->? Unit

Tree

: Type+ -> Type

%[free type Tree(a : Type+)

::= Leaf : Tree a_v-1

Branch : Tree a_v-1 * Tree a_v-1 -> Tree a_v-1

(head : Tree a_v-1; tail : Tree a_v-1)]%

Tree1

: Type+ -> Type

%[generated type Tree1(a : Type+)

::= Leaf : Tree1 a_v-1

Branch : Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1

(head : Tree a_v-1; tail : Tree1 a_v-1)]%

Tree2

: Type+ -> Type

%[type Tree2(a : Type+)

::= Leaf : Tree2 a_v-1

Branch : Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1

(head : Tree a_v-1; tail : Tree2 a_v-1)]%

Unit : Type := Unit

__*__ : Type+ -> Type+ -> Type

__-->__ : Type- -> Type+ -> Type

__-->?__ : Type- -> Type+ -> Type

__->__ : Type- -> Type+ -> Type

__->?__ : Type- -> Type+ -> Type

a : Type+ %(var_2)%

b : Type %(var_1)%

even

: Type+ -> Type

%[free type even(a : Type+)

::= rek : odd a_v-1 -> even a_v-1 (odd a_v-1)]%

even2 : Type+ -> Type %(data type)%

odd

: Type+ -> Type

%[free type odd(a : Type+)

::= rek : even a_v-1 -> odd a_v-1 (even a_v-1)]%

odd2

: Type+ -> Type

%[free type odd2(a : Type+)

::= rek : even2 a_v-1 -> odd2 a_v-1 (even2 a_v-1)]%

%% Assumptions -----------------------------------------------------------

Branch

: forall a : Type+ . Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1

%(construct Tree2)%

: forall a : Type+ . Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1

%(construct Tree1)%

: forall a : Type+ . Tree a_v-1 * Tree a_v-1 -> Tree a_v-1

%(construct Tree)%

Cons

: forall a : Type+ . a_v-1 * List a_v-1 -> List a_v-1

%(construct List)%

Leaf

: forall a : Type+ . Tree2 a_v-1 %(construct Tree2)%

: forall a : Type+ . Tree1 a_v-1 %(construct Tree1)%

: forall a : Type+ . Tree a_v-1 %(construct Tree)%

Nil : forall a : Type+ . List a_v-1 %(construct List)%

__/\__ : ? Unit * ? Unit ->? Unit %(fun)%

__<=>__ : ? Unit * ? Unit ->? Unit %(fun)%

__=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit %(fun)%

__=>__ : ? Unit * ? Unit ->? Unit %(fun)%

__=e=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit %(fun)%

__\/__ : ? Unit * ? Unit ->? Unit %(fun)%

__if__ : ? Unit * ? Unit ->? Unit %(fun)%

__when__else__

: forall a : Type . a_v-1 * ? Unit * a_v-1 ->? a_v-1 %(fun)%

bottom : forall a : Type . a_v-1 %(fun)%

def__ : forall a : Type . a_v-1 ->? Unit %(fun)%

false : Unit %(fun)%

head

: forall a : Type+ . Tree2 a_v-1 -> Tree a_v-1

%(select from Tree2 constructed by

Branch : forall a : Type+ .

Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1)%

: forall a : Type+ . Tree1 a_v-1 -> Tree a_v-1

%(select from Tree1 constructed by

Branch : forall a : Type+ .

Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1)%

: forall a : Type+ . Tree a_v-1 -> Tree a_v-1

%(select from Tree constructed by

Branch : forall a : Type+ . Tree a_v-1 * Tree a_v-1 -> Tree a_v-1)%

: forall a : Type+ . List a_v-1 -> a_v-1

%(select from List constructed by

Cons : forall a : Type+ . a_v-1 * List a_v-1 -> List a_v-1)%

not__ : ? Unit ->? Unit %(fun)%

rek

: forall a : Type+ . even2 a_v-1 -> odd2 a_v-1 %(construct odd2)%

: forall a : Type+ . even a_v-1 -> odd a_v-1 %(construct odd)%

: forall a : Type+ . odd a_v-1 -> even a_v-1 %(construct even)%

tail

: forall a : Type+ . Tree2 a_v-1 -> Tree2 a_v-1

%(select from Tree2 constructed by

Branch : forall a : Type+ .

Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1)%

: forall a : Type+ . Tree1 a_v-1 -> Tree1 a_v-1

%(select from Tree1 constructed by

Branch : forall a : Type+ .

Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1)%

: forall a : Type+ . Tree a_v-1 -> Tree a_v-1

%(select from Tree constructed by

Branch : forall a : Type+ . Tree a_v-1 * Tree a_v-1 -> Tree a_v-1)%

: forall a : Type+ . List a_v-1 -> List a_v-1

%(select from List constructed by

Cons : forall a : Type+ . a_v-1 * List a_v-1 -> List a_v-1)%

true : Unit %(fun)%

�__ : ? Unit ->? Unit %(fun)%

%% Sentences -------------------------------------------------------------

forall a : Type+; x_1_1 : a_v-1; x_1_2 : List a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op head : forall a : Type+ . List a_v-1 -> a_v-1)

((op Cons : forall a : Type+ . a_v-1 * List a_v-1 -> List a_v-1)

(var x_1_1 : a_v-1, var x_1_2 : List a_v-1)),

var x_1_1 : a_v-1) %(ga_select_head)%

forall a : Type+; x_1_1 : a_v-1; x_1_2 : List a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op tail : forall a : Type+ . List a_v-1 -> List a_v-1)

((op Cons : forall a : Type+ . a_v-1 * List a_v-1 -> List a_v-1)

(var x_1_1 : a_v-1, var x_1_2 : List a_v-1)),

var x_1_2 : List a_v-1) %(ga_select_tail)%

free type List(a : Type+)

::= Nil : List a_v-1

Cons : a_v-1 * List a_v-1 -> List a_v-1

(head : a_v-1; tail : List a_v-1) %(ga_List)%

free type List(a : Type+)

::= Nil : List a_v-1

Cons : a_v-1 * List a_v-1 -> List a_v-1

(head : a_v-1; tail : List a_v-1) %(ga_List)%

free type List(a : Type+)

::= Nil : List a_v-1

Cons : a_v-1 * List a_v-1 -> List a_v-1

(head : a_v-1; tail : List a_v-1) %(ga_List)%

free type List(a : Type+)

::= Nil : List a_v-1

Cons : a_v-1 * List a_v-1 -> List a_v-1

(head : a_v-1; tail : List a_v-1) %(ga_List)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op head : forall a : Type+ . Tree a_v-1 -> Tree a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree a_v-1 -> Tree a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree a_v-1)),

var x_1_1 : Tree a_v-1) %(ga_select_head)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op tail : forall a : Type+ . Tree a_v-1 -> Tree a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree a_v-1 -> Tree a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree a_v-1)),

var x_1_2 : Tree a_v-1) %(ga_select_tail)%

free type Tree(a : Type+)

::= Leaf : Tree a_v-1

Branch : Tree a_v-1 * Tree a_v-1 -> Tree a_v-1

(head : Tree a_v-1; tail : Tree a_v-1) %(ga_Tree)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree1 a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op head : forall a : Type+ . Tree1 a_v-1 -> Tree a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree1 a_v-1)),

var x_1_1 : Tree a_v-1) %(ga_select_head)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree1 a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op tail : forall a : Type+ . Tree1 a_v-1 -> Tree1 a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree1 a_v-1)),

var x_1_2 : Tree1 a_v-1) %(ga_select_tail)%

generated type Tree1(a : Type+)

::= Leaf : Tree1 a_v-1

Branch : Tree a_v-1 * Tree1 a_v-1 -> Tree1 a_v-1

(head : Tree a_v-1; tail : Tree1 a_v-1) %(ga_Tree1)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree2 a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op head : forall a : Type+ . Tree2 a_v-1 -> Tree a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree2 a_v-1)),

var x_1_1 : Tree a_v-1) %(ga_select_head)%

forall a : Type+; x_1_1 : Tree a_v-1; x_1_2 : Tree2 a_v-1

. (fun __=__ : forall a : Type . a_v-1 * a_v-1 ->? Unit)

((op tail : forall a : Type+ . Tree2 a_v-1 -> Tree2 a_v-1)

((op Branch

: forall a : Type+ . Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1)

(var x_1_1 : Tree a_v-1, var x_1_2 : Tree2 a_v-1)),

var x_1_2 : Tree2 a_v-1) %(ga_select_tail)%

type Tree2(a : Type+)

::= Leaf : Tree2 a_v-1

Branch : Tree a_v-1 * Tree2 a_v-1 -> Tree2 a_v-1

(head : Tree a_v-1; tail : Tree2 a_v-1) %(ga_Tree2)%

free type even(a : Type+)

::= rek : odd a_v-1 -> even a_v-1 (odd a_v-1)

free type odd(a : Type+)

::= rek : even a_v-1 -> odd a_v-1 (even a_v-1) %(ga_even_odd)%

free type odd2(a : Type+)

::= rek : even2 a_v-1 -> odd2 a_v-1 (even2 a_v-1) %(ga_even2_odd2)%

%% Diagnostics -----------------------------------------------------------

*** Hint 1.5, is type variable 'b'

*** Hint 2.5, is type variable 'a'

*** Hint 4.11, redeclared type 'List'

*** Error 5.50, unbound type variable(s)

b : Type in 'List b_v1'

*** Error 6.57, unexpected type argument 'b_v1'

*** Error 7.50, illegal polymorphic recursion

expected: List a_v-1

found: List (List a_v-1)

*** Error 14.27, illegal polymorphic recursion

expected: odd2 a_v-1

found: odd2 (odd2 a_v-1)