### Table 2. Rewriting rules on proof-nets for cut-elimination

"... In PAGE 3: ... Moreover, this rewriting enjoys strong normalisation and confluence [Ret93]. Table2 shows the rewriting rules on proof-nets. 1 a path of even length, starting and ending on the same vertex, using only once every other... ..."

Cited by 1

### Table 1: The Chemistry / Proof-Theory Roadmap (continued)

"... In PAGE 40: ...Table1 : The Chemistry / Proof-Theory Roadmap Chemistry -calculus Linear logic molecule as physical object (the nature of atoms and their bonds) -term in normal form proof-net in normal form (multiplicative fragment of lin- ear logic) shape as action (domain of interaction) (with whom and how a molecule interacts, as deter- mined by its shape and the na- ture of its reactive groups) type (speci cation of an action: a description of what the term does at a particular level of res- olution) multiset of propositional formulae (theorem) (a set of \actionable quot; ad- dresses, \interfaces quot;, or \plug- ging speci cations quot;) bonds (connectors of molecular parts) abstraction and application left and right rules of logical connectives initiation of a reaction application rule of inference: cut (reactants are the premises, products are the conclusions) completion of a reaction (structural rearrangements into stable products) normalization cut elimination branching reaction (multiple reaction pathways among the reactants) | multiple cut options (a sequent is a multiset) synthesis pathway (representation of a molecule as a suite of reactions between intermediates which enable its synthesis) -term containing redexes proof-net with cut(s) (a proof making use of inter- mediate results { lemmas a.k.... ..."

### Table 1: The Chemistry / Proof-Theory Roadmap (continued)

"... In PAGE 40: ...Table1 : The Chemistry / Proof-Theory Roadmap Chemistry -calculus Linear logic molecule as physical object (the nature of atoms and their bonds) -term in normal form proof-net in normal form (multiplicative fragment of lin- ear logic) shape as action (domain of interaction) (with whom and how a molecule interacts, as deter- mined by its shape and the na- ture of its reactive groups) type (speci cation of an action: a description of what the term does at a particular level of res- olution) multiset of propositional formulae (theorem) (a set of \actionable quot; ad- dresses, \interfaces quot;, or \plug- ging speci cations quot;) bonds (connectors of molecular parts) abstraction and application left and right rules of logical connectives initiation of a reaction application rule of inference: cut (reactants are the premises, products are the conclusions) completion of a reaction (structural rearrangements into stable products) normalization cut elimination branching reaction (multiple reaction pathways among the reactants) | multiple cut options (a sequent is a multiset) synthesis pathway (representation of a molecule as a suite of reactions between intermediates which enable its synthesis) -term containing redexes proof-net with cut(s) (a proof making use of inter- mediate results { lemmas a.k.... ..."

### Table 2: Proof of Theorem 3.2.

"... In PAGE 5: ... Let c S;#0F and c X;#0F refer to the closest gridpoint functions of c S and c X , respectively. We now explain the chain of inequalities as shown in Table2 needed for the proof. Note that by the sample size given in the statement of the theorem wehave uniform convergence for each f 2 F #0F 6 .... In PAGE 5: ... Thus the sample and true costs for each gridpointor#0F-net clustering are close. This implies that the values #282#29 and #283#29 as well as the values #285#29 and #286#29 in Table2 are close. Further, the #28sample or true#29 cost of any clustering and its nearest gridpoint clustering is no more than #0F 6 , hence the values #281#29 and #282#29 as well as the values #283#29 and #284#29, as well as the values #286#29 and #287#29 are close.... ..."

### Table 1: The Chemistry n2f Proof-Theory Roadmap n28continuedn29

"... In PAGE 43: ...Table1 : The Chemistry n2f Proof-Theory Roadmap Chemistry n15-calculus Linear logic molecule as physical object n28the nature of atoms and their bondsn29 n15-term in normal form proof-net in normal form n28multiplicative fragment of lin- ear logicn29 shape as action n28domain of interactionn29 n28with whom and how a molecule interacts, as deter- mined by its shape and the na- ture of its reactive groupsn29 type n28specin0ccation of an action: a description of what the term does at a particularlevel of res- olutionn29 multiset of propositional formulae n28theoremn29 n28a set of n5cactionable quot; ad- dresses, n5cinterfaces quot;, or n5cplug- ging specin0ccations quot;n29 bonds n28connectorsof molecularpartsn29 abstraction and application left and right rules of logical connectives initiation of a reaction application rule of inference: cut n28reactants are the premises, products are the conclusionsn29 completion of a reaction n28structural rearrangements into stable productsn29 normalization cut elimination branching reaction n28multiple reaction pathways among the reactantsn29 | multiple cut options n28a sequentisamultisetn29 synthesis pathway n28representation of a molecule as a suite of reactions between intermediates which enable its synthesisn29 n15-term containing redexes proof-net with cutn28sn29 n28a proof making use of inter- mediate results n7b lemmas a.k.... ..."

### Table 1: Selected results from LPO-Experiments Results for the KBO are very similar. For the second experiment, the net- work typically achieved more than 99% correctness on the test set. 4.3 Experiments in Fact Classi cation These experiments used only the simple labels from Sec. 4.1. The data was generated from PCL listings of DISCOUNT proof runs (compare [DS96b]). For each of 29 successful proof attempts, the generated equations were marked as either contributing to the proof or as direct derivatives of contributing facts. All other equations were discarded. We performed two di erent experiments. 7

"... In PAGE 7: ... The choice of the labeling scheme had only very minor in uence. Table1 shows selected results and the percentage of examples in the largest class of the test set { i.e.... ..."

### lable. Proof:

2000

Cited by 1

### Table 2. Proof System

2004

"... In PAGE 8: ... We now see that it also forms the basis of a sound and complete proof theory, and a decision procedure based on proof-search. The rules of the proof system are shown in Table2 . Since there is no Cut rule, the rules have a rather odd form.... ..."

Cited by 34