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### Proofs that count

Azadeh Farzan, Zachary Kincaid, Andreas Podelski
2014 Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14
Counting arguments are among the most basic proof methods in mathematics.  ...  While counting arguments are common in informal, hand-written proofs of such programs, there are no fully automated techniques to construct counting arguments.  ...  Recall that our use of counting proofs is the following (informal) proof rule: Suppose that P is a program, A, ϕ is a counting proof which satisfies the specification ψpre/ψpost, and that every trace of  ...

### Vote Counting as Mathematical Proof [chapter]

Dirk Pattinson, Carsten Schürmann
2015 Lecture Notes in Computer Science
Trust in the correctness of an election outcome requires proof of the correctness of vote counting.  ...  By formalising particular voting protocols as rules, correctness of vote counting amounts to verifying that all rules have been applied correctly.  ...  By analogy, a tree or sequence of correctly applied vote counting rules that determine the outcome of an election is a proof of the correctness of the count.  ...

### Logic and Complexity: Independence results and the complexity of propositional calculus [chapter]

Pavel Pudlák
1995 Proceedings of the International Congress of Mathematicians
The counting principle Count q is the statement that the set {0,... ,n -1} cannot be decomposed into blocks of size q, for every n not divisible by q.  ...  . , # m are a proof that (1) does not have a solution. Hubert's Nullstellensatz is a completeness result for this "calculus", it says that there is such a proof if (1) is unsolvable.  ...

### System Description: CYNTHIA [chapter]

Jon Whittle, Alan Bundy, Richard Boulton, Helen Lowe
1999 Lecture Notes in Computer Science
Selecting this term produces the nal program: tree -> int fun count (leaf n) = 1 | count (node(x,xs,ys)) = 1 + (count ys) + (count xs); Representing ML De nitions as Proofs Each ML function is represented  ...  In C Y NTHIA, each ML function de nition is represented as a proof of a speci cation of that function, using the idea of proofs-as-programs 2].  ...

### A Framework for Proving Contract-Equipped Classes [chapter]

Bertrand Meyer
2003 Lecture Notes in Computer Science
As part of a general effort to provide a new basis for software development through reuse of "Trusted Components", we outline a scheme for proving that classes equipped with contracts in the Eiffel style  ...  The approach takes advantage of the inheritance structure to separate proof obligations between deferred (abstract) classes, to be validated against a model, and their effective implementations, which  ...  The proof of the second postcondition clause, count = old count + 1, is similar, using the postcondition of extended in SEQUENCE.  ...

### Proof Complexity and the Kneser-Lovász Theorem [chapter]

2014 Lecture Notes in Computer Science
This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1).  ...  On the other hand for the cases k = 2, 3 (for which combinatorial proofs of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3) proofs.  ...  Ant 2,n ∧ Onto 2,n U r + P (2) r ≤ q r · (n − 1) has poly-size LK proofs. Proof. U r counts sets {i, j} such that both i and j are special.  ...

### Proving the monotonicity criterion for a plurality vote-counting program as a step towards verified vote-counting

Rajeev Gore, Thomas Meumann
2014 2014 6th International Conference on Electronic Voting: Verifying the Vote (EVOTE)
We show how modern interactive verification tools can be used to prove complex properties of vote-counting software.  ...  The command "detex evote14.tex | wc" indicates that our souce-file contains 4426 words, including the appendix.  ...  The biggest advantage of this method is that we can trust the final proof completely. The disadvantage is that the user has to be expert in logic and formal proof. III.  ...

### Graph-Based Proof Counting and Enumeration with Applications for Program Fragment Synthesis [chapter]

J. B. Wells, Boris Yakobowski
2005 Lecture Notes in Computer Science
We formally study proof enumeration and counting in this calculus. We prove that proof counting is solvable and give an algorithm to solve it. This in turn yields a proof enumeration algorithm.  ...  To solve the problem, we use the Curry-Howard correspondence (propositions-as-types, proofs-as-programs) to transform it into a proof enumeration problem for an intuitionistic logic calculus.  ...  We believe our approach to proof counting and enumeration is the first that has the following properties.  ...

### Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Samuel R. Buss, Leszek Aleksander Kołodziejczyk, Konrad Zdanowski
2015 Transactions of the American Mathematical Society
Recall: The "quasipolynomial simulation" means there is a 2 log O(1) n -time procedure to thusly convert AC 0 [p k ] proofs. Open: Does this hold for composite m as well?  ...  Collapsing Modular Counting Propositional proof systems Constant depth proofs Constant depth PK ⊕ p and PCK i p Constant depth PK ⊕p proofs allow ∧'s, ∨'s, and ⊕ p gates to appear at any level.  ...  Jeřábek showed that APC 1 can count the size of polynomial time sets to within a constant fraction ǫ. Namely, let X , Y ⊆ 2 n be defined by Boolean circuits.  ...

### Proof Complexity and the Kneser-Lovász Theorem [article]

2018 arXiv   pre-print
This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k=1).  ...  On the other hand for the cases k=2,3 (for which combinatorial proofs of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k=2), respectively extended Frege (k=3) proofs.  ...  Ant 2,n ∧ Onto 2,n U r + P (2) r ≤ q r · (n − 1) has poly-size LK proofs. Proof. U r counts sets {i, j} such that both i and j are special.  ...

### Page 753 of The Insurance Law Journal Vol. 6, Issue 10 [page]

1877 The Insurance Law Journal
But the plaintiff insists that this first count of the declaration was not intended as a special count on the policy, but as a count based upon the ad- justment only, and that the setting forth of the  ...  regulating this proof.  ...

### Starling: Lightweight Concurrency Verification with Views [chapter]

Matt Windsor, Mike Dodds, Ben Simner, Matthew J. Parkinson
2017 Lecture Notes in Computer Science
Starling takes a proof outline written in an abstracted Hoare-logic style, and converts it into proof terms that can be discharged by a sequential solver.  ...  Shared-variable version of ARC, and proof.  ...  The final important properties represented in the proof are, first, that the ARC is not disposed until all references are removed; and, second, that count accurately records the number of references.  ...

### Lower Bounds on Hilbert's Nullstellensatz and Propositional Proofs

Paul Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, Pavel Pudlák
1996 Proceedings of the London Mathematical Society
The weak form of the Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q;(%) = 0, does not have a solution in the algebraic closure 8 1 is in the ideal generated by the  ...  Denote by FP the eflect of ap-Proof: Assume that II is a depth d , size N k ( N ) Frege proof of Count: from instances of some Count? (and thus Mi = automatically.)  ...  ) = br(Tg)3. i f 4 = Count: E r then brl(T+) = 0 # br(T+) Assume now that II is a short constant-depth F'rege proof of Count: from some instances of CountR; that is from some formulas where M E j (mod  ...

### Page 4 of The Insurance Law Journal Vol. , Issue 300 [page]

1948 The Insurance Law Journal
as to that count.  ...  Proof of Loss Regarding the count in the declaration on the policy, there is authority that under the facts stated the proof of loss was waived by the insurer for two reasons: (1) “Proof of loss is waived  ...

### Improved estimate for the prime counting function π(x) [article]

Theophilus Agama
2018 arXiv   pre-print
Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions.  ...  In particular we show that π(x)=Θ(x)+O(1/ x), where Θ(x)=θ(x)/ x+x/2 x-1/4- 2/ x∑_n≤ x Ω(n)=k k≥ 2 2| n (x/n)/ 2. This is an improvement to the estimate π(x)=θ(x)/ x+O(x/^2 x) found in the literature.  ...  There exist some constant c > 0 such that θ(x) = x + O x e c √ log x . Proof. For a proof, see for instance  . Lemma 3.2. For all x ≥ 2 π(x) = θ(x) log x + x 2 θ(t) t log 2 t dt. Proof.  ...
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