Algebra of Proofs by M. E. Szabo

By M. E. Szabo

The following we learn the algebraic houses of the facts concept of intuitionist first-order common sense in a express environment. Our paintings relies at the confluence of rules and methods from evidence concept, class conception, and combinatory good judgment, and this e-book is addressed to experts in all 3 areas.Proof theorists will locate that different types provide upward push to a non-trivial semantics for evidence thought during which the concept that of the equivalence of proofs will be investigated from a mathematical standpoint. Categorists, however, will locate that evidence idea presents an appropriate syntax within which commutative diagrams may be characterised and labeled successfully. staff in combinatory good judgment, eventually, may well derive new insights from the learn of algebraic invariance houses in their thoughts verified during our presentation.

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6) Fm(H)(a(A, B, C)) = a(Fm(N)(A), Fm(H)(B), Fm(H)(C)) for all A, B, C E ObFm(C). (7) Fm(H)(A(A)) = A(Fm(H)(A)) for all A E ObFm(C). (8) Fm(H)(p(A)) = p(Frn(H)(A)) for all A E ObFrn(C). (9) Fm(H)(comp(g,f 1) = cornp(Fm(H)(I:),Fm(H)(f))for all c 0 r n p k - f ) E ArFm(C). (10) Fm(H)(f x1 g ) = Fm(H)(f) x1 Fm(H)(g) for all f. g E ArFm(C). The verification that Um and F m are adjoint functors is routine. We now show that there exists an alternative composition-free description of Fm(X) by means of an unlabelled deductive system mA(X).

19) comp(c+,a)= l(dom(a)). We call the category Fsm(X) the free symmetric monoidal category generated by X. 6, with the following additional clause: (1 1) Fsm(H)(a(A, B)) = a(Fsm(H)(A), Fsm(H)(B)) for all A, B E ObFsm(X). As in Chapter 2, the verification that Usm and Fsm are adjoint functors is routine. We now extend the composition-free description of Fm(X) to a composition-free description of Fsm(X). 4. 1. The deductive system smA(X) results from mA(X) by the inclusion of Rule (R4) as additional rule of inference.

THE CHURCH-ROSSER THEOREMFOR smA(X). If f = g , then there exists a normal h E Der(smA(X)) such thaf f 2 h and g 2 h. 61 THE S Y N T A X OF Fsm(X) 37 is sufficient to show that distinct normal derivations f, g : A + a represent distinct arrows in Ens. 1, we may assume that X is discrete. 3). 6) that f and g contain no instances of (R2). 1, f quotes an axiom iff g quotes an axiom. Under these conditions, neither f nor g contains an instance of (R4). 26), the same is true if both f and g end with an instance of (R8).

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