# Complexity of Proofs and Their Transformations in Axiomatic by V. P. Orevkov

By V. P. Orevkov

The purpose of this paintings is to increase the software of logical deduction schemata and use it to set up top and decrease bounds at the complexity of proofs and their adjustments in axiomatized theories. the most effects are institution of higher bounds at the elongation of deductions in minimize eliminations; an explanation that the size of a right away deduction of an life theorem within the predicate calculus can't be bounded above through an undemanding functionality of the size of an oblique deduction of a similar theorem; a complexity model of the lifestyles estate of the confident predicate calculus; and, for sure formal structures of mathematics, regulations at the complexity of deductions that make sure that the deducibility of a formulation for all usual numbers in a few finite set implies the deducibility of an analogous formulation with a common quantifier over all sufficiently huge numbers.

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We will say that v is a-hanging in r if v is a-hanging in r relative to the list of all function symbols of F'. We will say that an a-unknown v is /3-hanging in r if v is minimal in r and at least one of the following conditions holds: (1) there are r-atoms A and B such that A and B begin with different symbols, v r A and v r B ; (2) there are a r-term Q, r-atoms A and B, and a function symbol or object variable g such that A begins with g, A tir v, v tir B, B r Q, and g Q belongs to the list of constraints of F.

4, one easily shows that the succedents of the and 2 coincide with the succedents of end-sequents of 2, 1, , respectively. We now apply the end-sequents of and X2,1 , the inductive hypothesis to the premises of L* in (5). 10 and the inductive hypothesis, the result is the required proof. 2. 1 1 - 1 V -p, or -+ El. Let l = 1 if L" is an V2, & , & V1 , or V -p, and l = 2 if L" is an application application of & - 1 , & is of -+ 1, -+ V 1, - V2 , or -+ 2. In this case the list X3_1 is empty, empty if l = 1, and U is empty if l = 2.

L deg [ , ] L deg° [ , ] (5) g'' is q+-embeddable in J . PROOF. 4, we rebuild as a pure variable proof of S. 2. 9). 3 is used to obtain inequalities (3) and (4). 1. For any q+-closed set of formulas V. any proof in KGL(Qt) or in IGL(2() of a Q(-pure sequent S can be rebuilt as a proof ' such that (1) ' is a proof of the same sequent S in the same calculus as (2) gy' contains no D-applications of the cut rule; (3) h[7'] 2V §5. THE CALCULI KH(2t) AND IH(2t) 21[x]-1 (4) h° g"' <2 - 10 +1 <- l[am] 25 .