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Data Eq (A:*)(a:A) : (b:A)* where refl : Eq A a a;
repl : (A:*)(a:A)(b:A)(q:Eq _ a b)(P:(a:A)*)(p:P a)(P b);
intros;
induction q;
fill p;
Qed;
Freeze repl;
trans : (A:*)(a:A)(b:A)(c:A)(p:Eq _ a b)(q:Eq _ b c)(Eq _ a c);
intros;
induction q;
fill p;
Qed;
Freeze trans;
sym : (A:*)(a:A)(b:A)(p:Eq _ a b)(Eq _ b a);
intros;
induction p;
refine refl;
Qed;
Freeze sym;
Repl Eq repl sym;
eq_resp_f:(A,B:*)(f:(a:A)B)(x:A)(y:A)(q:Eq _ x y)(Eq _ (f x) (f y));
intros;
induction q;
refine refl;
Qed;
Freeze eq_resp_f;
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