Formula together with its polarity in a sequent, which is true if it is in the antecedent.
Decodes a term into its corresponding instance.
Decodes a term into its corresponding instance.
The resulting instance can contain alpha in the inductive case.
Encodes an expansion sequent (of an instance proof).
Encodes an expansion sequent (of an instance proof).
The shallow formulas of the expansion sequents should be subsumed by formulas in the end-sequent.
Encodes a sequent consisting of instances of an instance sequent.
Maps a function symbol to its corresponding formula in the end-sequent.
Assigns each formula in the end-sequent a fresh function symbol used to encode its instances.
Assigns each formula in the end-sequent a fresh function symbol used to encode its instances.
Encodes instances of a prenex FOL Sigma_1 end-sequent as FOL terms.
In the case of cut-introduction, the end-sequent has no free variables and we're encoding a Herbrand sequent as a set of terms. A term r_i(t_1,...,t_n) encodes an instance of the formula "forall x_1 ... x_n, phi(x_1,...,x_n)" using the instances (t_1,...,t_n).
In the case of simple inductive proofs, the end-sequent contains one free variable (alpha). Here, we consider proofs of instance sequents, which are obtained by substituting a numeral for alpha. Hence the formulas occuring in the end-sequents of instance proofs are substitution instances of endSequent; the encoded terms still only capture the instances used in the instance proofs--i.e. not alpha.