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Inductive List (A : Set) : Set :=
| Nil : List A
| Cons : A -> List A -> List A.
Inductive eqlong : List nat -> List nat -> Prop :=
| eql_cons :
forall (n m : nat) (x y : List nat),
eqlong x y -> eqlong (Cons nat n x) (Cons nat m y)
| eql_nil : eqlong (Nil nat) (Nil nat).
Parameter V1 : eqlong (Nil nat) (Nil nat) \/ ~ eqlong (Nil nat) (Nil nat).
Parameter
V2 :
forall (a : nat) (x : List nat),
eqlong (Nil nat) (Cons nat a x) \/ ~ eqlong (Nil nat) (Cons nat a x).
Parameter
V3 :
forall (a : nat) (x : List nat),
eqlong (Cons nat a x) (Nil nat) \/ ~ eqlong (Cons nat a x) (Nil nat).
Parameter
V4 :
forall (a : nat) (x : List nat) (b : nat) (y : List nat),
eqlong (Cons nat a x) (Cons nat b y) \/
~ eqlong (Cons nat a x) (Cons nat b y).
Parameter
nff :
forall (n m : nat) (x y : List nat),
~ eqlong x y -> ~ eqlong (Cons nat n x) (Cons nat m y).
Parameter
inv_r : forall (n : nat) (x : List nat), ~ eqlong (Nil nat) (Cons nat n x).
Parameter
inv_l : forall (n : nat) (x : List nat), ~ eqlong (Cons nat n x) (Nil nat).
Fixpoint eqlongdec (x y : List nat) {struct x} :
eqlong x y \/ ~ eqlong x y :=
match x, y return (eqlong x y \/ ~ eqlong x y) with
| Nil, Nil => or_introl (~ eqlong (Nil nat) (Nil nat)) eql_nil
| Nil, Cons a x as L => or_intror (eqlong (Nil nat) L) (inv_r a x)
| Cons a x as L, Nil => or_intror (eqlong L (Nil nat)) (inv_l a x)
| Cons a x as L1, Cons b y as L2 =>
match eqlongdec x y return (eqlong L1 L2 \/ ~ eqlong L1 L2) with
| or_introl h => or_introl (~ eqlong L1 L2) (eql_cons a b x y h)
| or_intror h => or_intror (eqlong L1 L2) (nff a b x y h)
end
end.
Type
match Nil nat as x, Nil nat as y return (eqlong x y \/ ~ eqlong x y) with
| Nil, Nil => or_introl (~ eqlong (Nil nat) (Nil nat)) eql_nil
| Nil, Cons a x as L => or_intror (eqlong (Nil nat) L) (inv_r a x)
| Cons a x as L, Nil => or_intror (eqlong L (Nil nat)) (inv_l a x)
| Cons a x as L1, Cons b y as L2 =>
match eqlongdec x y return (eqlong L1 L2 \/ ~ eqlong L1 L2) with
| or_introl h => or_introl (~ eqlong L1 L2) (eql_cons a b x y h)
| or_intror h => or_intror (eqlong L1 L2) (nff a b x y h)
end
end.
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