∈Z E,1 ψ Moreover, the instances of the reduction procedure ▷∈Z for ∈Z I− and ∈Z E−rules is as below: D1 Zb ∈ b D2 ¬Zb ∈ Zb ∈Z IZb ∈ {x ∈ b|¬x ∈ x} [Zb ∈ b]1, [¬Zb ∈ Zb]1 D3 ψ ∈Z E,1 ψ ▷∈Z D1 Zb ∈ b, D2 ¬Zb ∈ Zb D3 ψ 124 Let S∈Z be a system having ∈Z − and ¬−rules. R∈Z be a set of reductions including ▷¬ and ▷∈Z . Then, we have the result below. Proposition 3.4.1. There is an open derivation of ⊥ from Zb ∈ b in S∈Z relative to R ∈Z which generates a non-terminating reduction sequence, and so is not fully normalizable. Proof. two claims verify the result. Claim 1. there is an open derivation Σ2 of ⊥ from the assumption [Zb ∈ b]. We begin with an open derivation Σ1 of ⊥ from [Zb ∈ Zb]. [Zb ∈ Zb]1************************** de f Zb ∈ {x ∈ b|¬x ∈ x} [¬Zb ∈ Zb]2 [Zb ∈ b]2 [¬Zb ∈ Zb]2 ∈Z IZb ∈ {x ∈ b|¬x ∈ x}************************** de f Zb ∈ Zb [⊥]3 ¬E,3⊥ ∈Z E,2⊥ Then, there is an open derivation Σ2 of Zb ∈ Zb from [Zb ∈ b]. [Zb ∈ b]4 [Zb ∈ Zb]1 Σ1 ⊥ ¬I,1¬Zb ∈ Zb ∈Z IZb ∈ {x ∈ b|¬x ∈ x}************************** de f Zb ∈ Zb Now, we have an open derivation Σ3 of ⊥ from [Zb ∈ b]. [Zb ∈ Zb]5 Σ1 ⊥ ¬I,5¬Zb ∈ Zb [Zb ∈ b]4 Σ2 Zb ∈ Zb [⊥]6 ¬E,6⊥ Claim 2. Σ3 initiates a non-terminating reduction sequence and so is not fully normalizable. 125 By applying ▷¬ to Σ3, we have the derivation Σ4 below. [Zb ∈ b]4 [Zb ∈ Zb]1 Σ1 ⊥ ¬I,1¬Zb ∈ Zb ∈Z IZb ∈ {x ∈ b|¬x ∈ x} [¬Zb ∈ Zb]2 [Zb ∈ b]2 [¬Zb ∈ Zb]2 ∈Z IZb ∈ {x ∈ b|¬x ∈ x}************************** de f Zb ∈ Zb [⊥]3 ¬E,3⊥ ∈Z E,2⊥ Since Zb ∈ {x ∈ b|¬x ∈ x} in ∈Z E−rule is not an assumption, the application of ▷∈Z− reduction produces the same derivation with Σ3. Therefore, Σ3 generates a non-terminating reduction sequence and so is not fully normalizable. We call the result of Proposition 3.4.1 Crabbé's case. Furthermore, from the derivation Σ3