By Klaus Aehlig, Jan Johannsen, Helmut Schwichtenberg (auth.), Reinhard Kahle, Peter Schroeder-Heister, Robert Stärk (eds.)
Proof idea has lengthy been validated as a easy self-discipline of mathematical common sense. It has lately turn into more and more appropriate to laptop technological know-how. The - ductive gear supplied through evidence thought has proved important for metatheoretical reasons in addition to for functional functions. therefore it looked as if it would us such a lot usual to carry researchers jointly to evaluate either the function facts concept already performs in laptop technological know-how and the position it could actually play sooner or later. the shape of a Dagstuhl seminar is best suited for reasons like this, as Schloß Dagstuhl offers a truly handy and stimulating setting to - scuss new principles and advancements. To accompany the convention with a proc- dings quantity looked as if it would us both acceptable. the sort of quantity not just ?xes simple result of the topic and makes them on hand to a broader viewers, but in addition indications to the scienti?c neighborhood that facts idea in computing device technology (PTCS) is a tremendous learn department in the wider ?eld of good judgment in computing device science.
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Extra resources for Proof Theory in Computer Science: International Seminar, PTCS 2001 Dagstuhl Castle, Germany, October 7–12, 2001 Proceedings
0]2+ + (0 + 0) . , the underlined) occurrence of 0 + 0. Note that r2p (0 + 0) ≡ r1p (0). The needed instance T of the transitivity axiom is (r1p (0 + 0) = r1p (0) ∧ r2p (0 + 0) = r2p (0)) ⊃ r1p (0 + 0) = r2p (0). Let and tp1 ≡ p+ tp2 ≡ p−1 + (. . + (2+ + 0) . ). + Note that tp2 ≡ r2p (0) and tp1 + tp2 ≡ r1 (0 + 0); tp1 is identical to the parameter term and the conclusion of the instance of the transitivity axiom is identical to instance of the premise in x + y = y ⊃ x = 0. Consequently the sequent is of form N ⊃ B1 , N ⊃ B2 , (B1 ∧ B2 ) ⊃ B ((B ⊃ C) ∧ N ) ⊃ C and therefore tautological.
Ferm¨ uller Proof. Let Therefore and r1p (z) ≡ f p−1 (z) + (. . + (z + 0) . ). r1p (f (0)) ≡ f p (0) + (. . + (f (0) + 0) . ) r1p (0) ≡ f p−1 (0) + (. . + (0 + 0) . ) Note that this implies that also the leftmost formula of the sequent is an instance of ID= . Let r2p (z) ≡ f p−1 (0) + (. . + (f (0) + z . )). Setting and tp1 ≡ f p (0) tp2 ≡ f p−1 (0) + (. . + (f (0) + 0) . ) Remark 4. Note that in the proof above two instances of the schema of identity are used. ) Again, one can show that the two instances of ID= cannot be replaced by a uniformly bounded number of instances of identity axioms: Proposition 3.
Every instance of the schema of identity is derivable from identity axioms. However, from a proof theoretic perspective the schema of identity is quite powerful. Even one instance of it can be used to prove formulas uniformly where no ﬁxed number of instances of identity axioms is suﬃcient for this purpose. 1 Introduction In this paper we investigate the impact of replacing instances of the schema of identity by instances of identity axioms in valid Herbrand disjunctions. More precisely, we consider valid sequents of form Ξ, s = t ⊃ (B(s) ⊃ B(t)) ∃xA(rp , x) where Ξ consists of identity axioms, A is quantiﬁer free, and rp is some “parameter term”, for which the index p can be thought of as its size.
Proof Theory in Computer Science: International Seminar, PTCS 2001 Dagstuhl Castle, Germany, October 7–12, 2001 Proceedings by Klaus Aehlig, Jan Johannsen, Helmut Schwichtenberg (auth.), Reinhard Kahle, Peter Schroeder-Heister, Robert Stärk (eds.)