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This paper presents two extensions of the second order polymorphic lambda calculus F with inductive and coinductive types including not only (co)iteration but primitive (co)recursion and inversion principles. The systems are proven to be safe, can be seen as extensions of Hagino's categorical lambda calculus and are also related with the systems of higher-order iterators of .doi:10.1051/ita/2009015 fatcat:5q7oeuxdxfesnc3w4x3dsxav4y