We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Gödel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of E-HA ω and E-PA ω , strengthening earlier results by Moerdijk and Palmgren, and Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To conclude the paper, we will point out some open problems and directions for future research, including some initial results on saturation principles.

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Sulphur removal in the ironmaking and oxygen steelmaking process is reviewed. A sulphur balance is made for the steelmaking process of Tata Steel IJmuiden, the Netherlands. There are four stages where sulphur can be removed: in the blast furnace (BF), during hot metal (HM) pretreatment, in the converter and during the secondary metallurgy (SM) treatment. For sulphur removal a low oxygen activity and a basic slag are required. In the BF typically 90% of the sulphur is removed; still, the HM contains about 0.03% of sulphur. Different HM desulphurisation processes are used worldwide. With co-injection or the Kanbara reactor, sulphur concentrations below 0.001% are reached. Basic slag helps desulphurisation in the converter. However, sulphur increase is not uncommon in the converter due to high oxygen activity and sulphur input via scrap and additions. For low sulphur concentrations SM desulphurisation, with a decreased oxygen activity and a basic slag, is always required.

In mechanically ventilated patients with and without COPD, a time constant can well be calculated from the expiratory flow-volume curve for the last 75% of tidal volume, gives a good estimation of respiratory mechanics, and is easy to obtain at the bedside.

We define a notion of weak ω-category internal to a model of Martin-Löf's type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower of iterated identity types over that type. We show that the ω-categories arising in this way are in fact ω-groupoids.does not commute on the nose, but only up to suitable termsThus, if we wish to view A as an honest groupoid, we must first quotient out the sets of elements p ∈ Id(a, b) by propositional equality. A more familiar instance of the same phenomenon occurs in constructing the fundamental groupoid of a space, where we must identify paths up to homotopy, and this suggests the following analogy: that types are like topological spaces, and propositional equality is like the homotopy relation. Using the machinery of abstract homotopy theory, this analogy has been given a precise form in [1], which constructs type-theoretic structures from homotopy-theoretic ones, and in [6], which does the converse. Mathematics Subject Classification 3B15, 18D05 (primary); 18D50 (secondary).The second-named author also acknowledges the support of a Research Fellowship of St John's College, Cambridge and a Marie Curie Intra-European Fellowship, Project No. 040802. s z z u u u u u u t 6 6 J J J J J J s z z u u u u u u t 6 6 J J J J J JThus, it suffices to show that this cone coincides with (13); which is to show that, for each 1 i k, we have r πi = r πi • r 0 . Now, observe that r πi is obtained as ((Γ * ) +1 ) πi , where (Γ * ) +1 is the globular context Γ 0 r0 z z t t t t t t t t t t r0r1 r0r1r2 6 6 J