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These are three books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity

==============================================

Update. Jan Krajíček wrote a new book on this topic recently. You can find the final manuscript of it here.

These are three books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity

==============================================

Update. Jan Krajíček wrote a new book on this topic recently. You can find the final manuscript of it here.

These are three books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity

These are three books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity

==============================================

Update. Jan Krajíček wrote a new book on this topic recently. You can find the final manuscript of it here.

These are two books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity

These are three books that I know:

1. Logical Foundations of Proof Complexity
2. Bounded Arithmetic, Propositional Logic and Complexity Theory
3. Logical Foundations of Mathematics and Computational Complexity, Proof complexity chapter

Also, I suggest you read the following papers:

1. The lengths of proofs
2. Bounded Arithmetic and Propositional Proof Complexity