A monograph of the Mollusca from the Great Oolite, chiefly by Morris J., Lycett J. PDF

By Morris J., Lycett J.

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Additional resources for A monograph of the Mollusca from the Great Oolite, chiefly from Minchinhampton and the Coast of Yorkshire. Part I. Univalves

Sample text

Bn , C} C ⊸ {D1 , . . , Dk } pure substitution (SUB*) A ⊸ {B1 , . . , Bn , D1 , . . , Dk } A ⊸ {B1 , . . , Bn } restriction (RES) A ∪ C ⊸ {B1 ∪ C, . . , Bn ∪ C} A ⊸ {B1 , . . , Bn } C ⊸ {D1 , . . , Dm } distribution (D) A ∪ C ⊸ {Bi ∪ Dj | 1 ≤ i ≤ n, 1 ≤ j ≤ m} cumulated contradiction (CONT*) E E A [ R]A ∪ [ R]B ⊸ {[ R](A ∪ B)} cumulated A ⊥ ⊸ {A} -propagation (AP*) Proof: The validity of these rules is proven by providing the DR proof trees that realize them. 1. cumulated premise extension Let C = {C1 , .

III) for all δ, δ¯ ∈ ∆ and R ∈ MR we have (δ, δ) The proof of Fact I will be done by inductively showing that ϕ(δ) is contained in all ∆(i) occurring − → in the construction of K (K) (and therefore also in ∆ = i∈N ∆(i) ). 4 We use the term model synonymously for binary power context family. 1 Definition and Deduction Calculus 43 • induction anchor: ϕ(δ) ∈ ∆(0) . − → norm − = ∅, we have that ϕ(δ) = {C ∈ FLE Since K respects K by assumption and [[⊥]]→ | K (0) δ |= C} ∈ ∆ . E Furthermore, notice that (δ1 , δ2 )IR R implies δ1 |= R.

Dk } A ⊸ {B1 , . . , Bn } -lifting (EL) [ R]A ⊸ {[ R]B1 , . . B} ⊸ {[ R](A ∪ {B})} -propagation (AP) A ⊸ {B1 , . . , Bn , C} -lifting (AL) [ R]A ⊸ {[ R]B1 , . . A | A ∈ A} and {⊥}, if ⊥ ∈ A { R. 2 E E [ R]A := ❑ Given a set K of cumulated clauses, we denote by DR(K) the smallest set of cumulated clauses containing K and closed under the derivation rules above. For k ∈ DR(K) we also write K ⊢ k. The rules ID, PE, CE and SUB have already been described in [Kr98] as deduction rules for cumulated clauses on arbitrary sets.

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A monograph of the Mollusca from the Great Oolite, chiefly from Minchinhampton and the Coast of Yorkshire. Part I. Univalves by Morris J., Lycett J.


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