Act06_Subst_and_Pf_Rules

Act06_Subst_and_Pf_Rules - Illinois Institute of Technology...

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CS 536 Activities: Syntactic Substitution; More Proof Rules Activity 6.1: Syntactic Substitution A. Why? To reason about assignments, we need to understand syntactic substitution. B. Outcomes At the end of this activity you should: Be able to do syntactic substitution. C. Questions 1. Calculate ( x+i b+c = 0 )[ i+ 1/ i ][ b+c / c ]. For Questions 2 – 8, let P be x . x < y x ² y+k , where x , y , and k range over integers. 2. What is P [ 5 / x ]? The same as P , because we’re trying to substitute for x , the quanti fi ed variable. 3. What is P [ 5 / y ]? x . x < 5 x ² 5+k 4. What is P [ 5 / z ]? Same as P because z doesn’t occur in P at all. 5. What is P [ (x+y)/2 / y ]? To avoid capture, we’ll replace x by z . ( x . x < y x ² y+k) [ (x+y)/2 / y ] is z .( (x < y x ² y+k) [ z / x ] [ (x+y)/2 / y ]) is z .( (z < y z ² y+k) [ (x+y)/2 / y ]) is z . z < (x+y)/2 z ² (x+y)/2+k 6. Let Q be y . k . P . What is Q [ e / v ] where v is any variable (including possibly x , y , and k ) and e is any e x pression?
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