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# Lab for Automated Reasoning and Analysis LARA

# Differences

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+ | ====== Lecture 8 Skeleton ====== | ||

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+ | ==== Substitution theorem ==== | ||

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+ | Recall the statement in the optional part of Homework 1. | ||

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+ | Review of | ||

+ | * semantics of first-order logic | ||

+ | * definition of capture-avoiding substitution and the set of free variables | ||

+ | * structural induction on algebraic data types and comparison to induction on natural numbers | ||

+ | * proof of substitution theorem using structural induction | ||

+ | * the case of a quantifier as a justification for the definition of capture-avoiding substitution | ||

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+ | ==== Quantifier elimination from Homework 2 ==== | ||

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+ | * transformation to conjunctions of literals | ||

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+ | * the idea of Fourier-Motzkin elimination | ||

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+ | ==== Resolution through an example ==== | ||

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+ | (ALL x. EX y. R(x,y)) & | ||

+ | (ALL x y z. R(x,y) --> R(x, plus(y,z))) & | ||

+ | (Ev(x) | Ev(plus(x,1))) --> | ||

+ | (ALL x. EX y. R(x,y) & E(y)) | ||

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+ | The intuitive infinite model. | ||

+ | Example finite model where this holds. | ||

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+ | Review of transformation to clauses, unification, resolution rule in this example. | ||

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+ | The meaning of completeness and soundness of resolution. | ||

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+ | ==== Homework 3 review ==== | ||

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+ | Galois connection. | ||

+ | Characterization of injective and surjective functions. | ||

+ | Two equivalent definitions of Galois connection. | ||