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Unformatted text preview: Four Howtos Jan van Eijck [email protected] January 11, 2006 Abstract How to Set Up a Proof? How to Read and Analyse a Proof? How to Transform a Definition into a Haskell Implementation? How to Read and Understand a Haskell Function? How to Set Up a Proof Advice from [ 1 ]: • Proofs are highly structured texts. Be aware of their structure. • When constructing proofs, use the following schema: Given: . . . To be proved: . . . Proof: . . . • Look up definitions of defined notions, and use these definitions to rewrite both Given and To be proved . • Make sure you have a sufficient supply of scrap paper, and make a fair copy of the endproduct. • Ask yourself two things: Is this correct? Can others read it? Proof Recipes: Subproofs Given: A , B , . . . To be proved: P Proof: . . . Suppose C . . . To be proved: Q Proof: . . . . . . Thus Q . . . Thus P Scope of Assumptions The purpose of ‘Suppose’ is to add a new given to the list of assumptions that may be used, but only for the duration of the subproof of which ‘Suppose’ is the head. If the current list of givens is P 1 , . . . , P n then ‘Suppose Q’ extends this list to P 1 , . . . , P n , Q . In general, inside a box, you can use all the givens and assumptions of all the including boxes. Thus, in the innermost box of the example, the givens are A, B, C . This illustrates the importance of indentation for keeping track of the ‘current box’. Attitude Grasp the importance of proper formatting. Use indentation to clarify the structure of your proofs. By getting it on paper in a structured way, you will clear up confusion in your mind. The two ways of encountering a logical symbol 1. The symbol can appear in the given , or in an assumption . 2. The symbol can appear in the statement that is to be proved . In the first case the rule to use is an elimination rule, in the second case an introduction rule. Elimination rules enable you to reduce a proof problem to a new, hope fully simpler, one. Introduction rules make clear how to prove a goal of a certain given shape. Introduction of an Implication Given: . . . To be proved: Φ ⇒ Ψ Proof: Suppose Φ To be proved: Ψ Proof: . . . Thus Φ ⇒ Ψ . This rule is called the Deduction Rule . Example Given: R is a transitive relation To be proved: ( x, y ) ∈ R ◦ R ⇒ ( x, y ) ∈ R Proof: Suppose ( x, y ) ∈ R ◦ R To be proved: ( x, y ) ∈ R Proof: From ( x, y ) ∈ R ◦ R : there is a z with ( x, z ) ∈ R , ( z, y ) ∈ R . From this, by transitivity of R , ( x, y ) ∈ R . Thus ( x, y ) ∈ R ◦ R ⇒ ( x, y ) ∈ R Use (Elimination) of an implication This rule is also called Modus Ponens . Given: Φ ⇒ Ψ , Φ Thus Ψ . Examples Given: x ∈ A ⇒ x ∈ B , x ∈ A ....
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 Spring '08
 WOUTERS
 Algebra

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