### Table 1. Corresponding replacement proofs

2002

"... In PAGE 11: ... The possible instantia- tions of these grounded atoms in which t is replaced are t = v; v = t; vRt; tRv; P(t) and h x: i(t); where h x: i(x) is ATx( 0) for some 0. In Table1 the application of the replacement rule is given on the left while the corresponding proof on the AT images of the formulas is on the right. The cases for P(t), t = v, Rtv and h x: i(t) are all by applications of Nom.... ..."

Cited by 8

### Table 5.3: Lines, not in V(z), containing p, corresponding to line modules Proof This is straightforward after (5.1.11). Remark 5.2.4 If NA is a net of quadrics, then for every p we have Qp = V(au + bv + z2) with au + bv 6 = 0: Hence if p is of the third kind, then Qp is not smooth at p, and one obtains an in nite family of lines through p which are not in V(z): Example 5.2.5 Let A be a 3-dimensional Sklyanin algebra. That is, PA is a smooth elliptic curve and is a translation of the form p = p + with 3 6 = 0. Then p_ = p + 2 . If 2 = 0 then every point of PA is of the third kind, and hence there will be a lot of line modules.

1996

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### 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 2. Proof verification

2003

"... In PAGE 5: ... Table 2. Proof verification Table2 gives data about proof verification. The Verification time column shows the time taken by Proof_verification2.... In PAGE 6: ... (For the rest of the conflict clauses we computed the number of resolutions exactly. So we believe the lower bounds shown in Table2 are close to the real sizes.) The Confl.... In PAGE 6: ... It is not hard to see that with the exception of a few instances conflict clause proofs are smaller than resolution graph ones. (In Table2 we estimate only the initial size of a resolution graph. That is we do not take into account that, as it was mentioned in Section 5, the size of the resolution proof grows during proof verification.... In PAGE 6: ...onfl. cl. proof size (in thou- sands of literals) Ra- tio % bounded model checking, SAT-2002 [18] fifo8_200 379,992 71,971 18 fifo8_300 987,840 118,132 11 fifo8_400 4,581,450 335,752 7 Table 3. Growth of resolution proof size The size of the largest proof of Table2 (formula 7pipe) was 257 Mbyte and so we were able to verify the proof on the computer with 640 Mbytes of memory. On the other hand, the corresponding resolution graph proof contained 435 million nodes and so the resolution graph would take more than 2 Gbytes of memory (assuming that on average one needs 5 digits to label a node of the resolution graph).... ..."

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### Table 2. Proof verification

"... In PAGE 5: ...44 103,556 41.5 Table2 gives data about proof verification. The Verification time column shows the time taken by Proof_verification2.... In PAGE 5: ... (For the rest of the conflict clauses we computed the number of resolutions exactly. So we believe the lower bounds shown in Table2 are close to the real sizes.) The Confl.... In PAGE 5: ...It is not hard to see that with the exception of a few instances conflict clause proofs are smaller than resolution graph ones. (In Table2... In PAGE 6: ... cl. proof size (in thou- sands of literals) Ra- tio % bounded model checking, SAT-2002 [18] fifo8_200 379,992 71,971 18 fifo8_300 987,840 118,132 11 fifo8_400 4,581,450 335,752 7 The size of the largest proof of Table2 (formula 7pipe) was 257 Mbyte and so we were able to verify the proof on the computer with 640 Mbytes of memory. On the other hand, the corresponding resolution graph proof contained 435 million nodes and so the resolution graph would take more than 2 Gbytes of memory (assuming that on average one needs 5 digits to label a node of the resolution graph).... ..."

### Table 2: the proof of the target problem.

"... In PAGE 4: ... Table 2 shows how dltk abs is re ned to the proof of dltk 2 . Proof lines of Table2 with integer labels are obtained from the corresponding proof lines of dltk abs . Boxed terms and sub-formulas are the result of instantiations.... ..."

### Table 1: Sequent proof rules and corresponding typing rules

1993

"... In PAGE 34: ...let M be P in N )l N[ ] Match(M; P) = ( P :J) of M is Q in N )l let J[ ] be Q in N [M jL N] )l M [M jR N] )l N Match(M; P1@P2) fails let M be P1@P2 in N )l let hM; Mi be hP1; P2i in N Match(M; P1@P2) fails ( P1@P2:J) of M is Q in N )l ( hP1; P2i:J) of hM; Mi is Q in N Match(M; P) fails M P ; M0 let M be P in N )l let M0 be P in N Match(M; P) fails M P ; M0 ( P :J) of M is Q in N )l ( P :J) of M0 is Q in N M P1 ; M0 inl(M) (P1j P2) ; inl(M0) M P2 ; M0 inr(M) (P1j P2) ; inr(M0) Match(M1; P1) fails M1 P1 ; M10 hM1; M2i hP1;P2i ; hM10; M2i Match(M2; P2) fails M2 P2 ; M20 hM1; M20i hP1;P2i ; hM1; M20i M )l M0 M P ; M0 Table1 0: Lazy evaluator in SOS semantics style Lemma 6.1 If M well-typed and not a lazy-canonical form then we have M )l N for some N.... In PAGE 38: ...2 and Ce ranges over eager canonical forms. Match(Ce; P) = let Ce be P in N )e N[ ] M )e M0 let M be P in N )e let M0 be P in N [M jL N] )e M [M jR N] )e N Match(Ce; P) = ( P :J) of Ce is Q in M )e let J[ ] be Q in M N )e N0 ( P :J) of N is Q in M )e ( P :J) of N0 is Q in M M )e M0 hM; Ni )e hM0; Ni M )e M0 hN; Mi )e hN; M0i M )e M0 inl(M) )e inl(M0) M )e M0 inr (M) )e inr(M0) Table1 1: Eager evaluator in SOS semantics style Well-typed closed terms can always be reduced to a eager-canonical forms with the eager evaluator appearing in gure 10. We de ne a pattern P to specify the type A if and only if P and A correspond to one of the following cases: and x satisfy any type A.... In PAGE 41: ... Of course lazy pattern-matching can o er such a rst-order algorithm8, see table 12. nat def= recX:1 + X zero def= foldX:1+X(inlnat(?)) succ(M) def= foldX:1+X(inr1(M)) zero j succ(P) def= fold((? j P)) inf def= hzero j succ(x); zero j succ(y)i: [[zero j zero] j [zero j succ(inf hx; yi)]] : nat nat ! nat Table1 2: inf in the typed pattern calculus However, Colson also shows that higher-order (Godel apos;s T) primitive recursion algorithms exist with this inten- sional behavior. It remains open then to nd better evidence for the intuition that the typed pattern calculus is intensionally more expressive.... ..."

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### Table 1 This completes the proof of necessity.

"... In PAGE 5: ...ase 3. c0 = 1, n is even. Then c1xy + y2 = (?c0)n + c0x2 = 2. Taking into account that x, y, and c1 are integers, one can list all their possible values (see Table1 ). In addition, Table 1 contains the values of g (t) (the minimal polynomial for over Q) and the corresponding values of and n.... In PAGE 5: ... Condition (i) is obviously su cient, since 2 (Z[ ]) if c0 = 1. As to condition (ii), one can directly check that it is su cient (see the values of in Table1 ). This completes the proof.... ..."

### Table 3: Proof results and times

"... In PAGE 11: ... 4.2 Running the Theorem Provers Table3 summarizes the results obtained from running the theorem provers on all proof obli- gations (except for the invalid obhgahons from rhe in-use-ycjkyj, pic@ by amp;%rent simplification levels. Each line in the table corresponds to the proof tasks originating fiom a specific safety policy (array, init, in-use, symm, and norm).... In PAGE 12: ... Here, all simplification steps are required to make the obligations go through the first-order ATPs. The results in Table3 also indicate there is no single best theorem prover. Even variants of the same prover can differ widely in their results.... ..."