### Table 1. Conjectures proved by reuse

"... In PAGE 5: ... Apart from initial proofs provided by the hu- man advisor in the \prove quot; step, none of these steps necessitates human support. Thus the proof shell from Figure 3 can be automatically reused for proving the step formulas apos;s i of the apparently di erent conjectures apos;i given in Table1 below. This table illustrates a typical session with the Plagiator-system: At the beginning of the ses- sion the human advisor submits statement apos; (in the rst row) and a proof p of apos; to the system.... ..."

### Table 3.5: Level of user assistance required for LP proofs of queries Four of the proofs needed no assistance from the user: plug-in match of Stack.push and Queue.enq with Q2, and plug-in post and specialized matches of Stack.top with Q3. Plug-in match of Stack.push with Q2 is the example shown in Figure 3.7; executing the statements in Figure 3.7 results in the response from LP that the match conjecture was proved by normal- ization; no user assistance was required. Generalized match of Stack.pop with Q6 is an example of a match that requires some user assistance to LP. The user must tell the prover to use induction in the proof, and then how to instantiate the existential variables. Figure 3.8 shows an LP-annotated script for this proof. The lines with boldface are user input; lt; gt; and [ ] are proof notes from LP; and % is the comment character. The line [ ] conjecture indicates that LP completed the proof. We classify the user assistance for this proof as simply guidance { telling LP what proof strategy

1996

Cited by 20

### Table 5: Valid Proof Expressions

1993

"... In PAGE 18: ... We write FV( ) for the set of free variables of the formulas apos;i in , and dom( ) for the set of i apos;s in . In Table5 we give rules for proving assertions of the form X; ` : apos;, which is to be read as \ is a valid proof of apos; with respect to the proof context (X; ). quot; The derivability of such an assertion means that a number of general rules are obeyed (e.... In PAGE 20: ... To encode natural deduction proofs as LF terms we assume that the LF variables include the rst-order variables and the occurrence markers. For each proof context (X; ) we inductively de ne a function quot;X; from valid proofs with respect to (X; ) (de ned in Table5 ) to LF terms as follows: quot;X; (hyp apos;( )) = quot;X; (raa apos;( )) = raa quot;X( apos;) quot;X; ( ) quot;X; (imp-i apos;; ( : )) = imp-i quot;X( apos;) quot;X( ) :true( quot;X( apos;)): quot;X;( ; : apos;)( ) quot;X; (all-ix; apos;( )) = all-i ( x: : quot;X;x( apos;)) ( x: : quot;(X;x); ( )) quot;X; (all-ex; apos;;t( )) = all-e ( x: : quot;X;x( apos;)) quot;X(t) quot;X; ( ) quot;X; (some-ex; apos;; ( 0; : )) = some-e ( x: : quot;X;x( apos;)) quot;X( ) quot;X; ( 0) ( x: : :true( quot;X;x( apos;)): quot;(X;x);( ; : apos;)( )) In the cases involving individual variables x and occurrence markers , it is assumed that x and are chosen to be the rst such variable or occurrence marker (in some standard enumeration) not occurring in X or , respectively.... ..."

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### Tables Search and prediction of patterns Conjectures formulation Use of school knowledge Conjectures validation Conjectures generalization Characterization of even numbers

### Table I: Conjectured values of (N), the largest t for which an N-point con guration on the sphere in 3 dimensions forms a spherical t-design. N (N) Proof Group Order Orbits (Description)

1996

Cited by 12

### Table 3: Selected conjectures from the TPTP set theory sample with the proof cost measured by the number of clauses processed by the prover.

"... In PAGE 9: ... The meaning of the symbols that we use is summarized in Ta- ble 2. The conjectures that were proved are shown in Table3 . The table also shows the results obtained when running the system with the proofs being measured by the number of processed clauses.... ..."

### Table I: (cont.) Conjectured values of (N), the largest t for which an N-point con guration on the sphere in 3 dimensions forms a spherical t-design. N (N) Proof Group Order Orbits (Description)

1996

Cited by 12

### Table 1, where apos;0 is speculated as a lemma when the conjecture apos; is to be proved by reuse with the Plagiator system, are compared by gt;F . Columns (a), (b), and (c) compare conjectures and lemmata by criteria (a), (b), and (c) from De nition 3.

1996

"... In PAGE 7: ...roof p0 of , cf. [13]. Apart from initial proofs provided by the human advisor in the \prove quot; step, none of these steps necessitates human support. Thus the proof shell from Figure 3 can be automatically reused for proving the step formulas apos;s i of the apparently di erent conjectures apos;i given in Table1 below ( lt; gt; abbreviates append). This table illustrates a typical session with the Plagiator-system: At the beginning of the session the human advisor submits statement apos; (in the rst row) and a proof p of apos; to the system.... In PAGE 7: ...ystem when proving a statement by reuse. E.g. statement apos;6 is speculated when verifying apos;2, which leads to speculating apos;7 which in turn entails speculation of conjecture apos;8, for which eventually apos;9 is speculated. apos; P x + Py P(x lt; gt; y) apos;9 F1(G1(x); G2(y)) G3(H1(x; y)) No: Conjectures proved by reuse Subgoals apos;0 j x j + j y j j x lt; gt; y j ? apos;1 2x + 2y 2(x + y) ? apos;2 (zy)x zx y apos;6 apos;3 Q x Q y Q(x lt; gt; y) apos;7 apos;4 x + y y + x apos;10 apos;5 reverse(y) lt; gt; reverse(x) reverse(x lt; gt; y) ( apos;11) apos;6 zx zy zx+y apos;7 apos;7 x (y z) (x y) z apos;8 apos;8 x z + y z (x + y) z apos;9 apos;9 x + (y + z) (x + y) + z ? apos;10 x + succ(y) succ(x + y) ? apos;11 x lt; gt; (y lt; gt; z) (x lt; gt; y) lt; gt; z ? Table1 . Conjectures proved by reuse 4 Reusing Proofs as Problem Reduction Our method for reusing proofs can be viewed as an instance of the problem re- duction paradigm, where a problem p is mapped to a nite set of subproblems fp1; :::; png by some (problem-)reduction operators, and each of the subproblems pi is mapped to a nite set of subproblems in turn, etc.... In PAGE 10: ... Theorem 2. For instance, reconsider the exam- ples from Table1 : Proving apos;3 by reuse leads to speculating the lemmata apos;7, apos;8, and apos;9 in turn. We nd purify S( apos;3) = fprod; appendg ftimesg = purify S( apos;7) = purify S( apos;8) fplusg = purify S( apos;9) and #times( apos;7) = 4 gt; 3 = #times( apos;8), and therefore apos;3 gt;F apos;7 gt;F apos;8 gt;F apos;9.... In PAGE 10: ... Since purify S( apos;2) = fexpg = purify S( apos;6) and #exp( apos;2) = 3 = #exp( apos;6), apos;2 gt;F apos;6 does not hold. Also apos;4 6 gt;F apos;10 for apos;4 := plus(x; y) plus(y; x) and apos;10 := plus(x; succ(y)) succ(plus(x; y)) from Table1 , because purify S( apos;4) = fplusg = purify S( apos;10) and #plus( apos;4) = 2 = #plus( apos;10). As a remedy, we also consider the arguments in an application of a maxi- mal function symbol in a conjecture.... ..."

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