Class 16: Structural Induction cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans Mohammad Mahmoody University of Virginia 

Plan for today •Continuing Recursive Data Types •Structural Induction 

Problem Set 6 Due tomorrow Problem 7 “correction” 

Defining List Definition 1. A List is an ordered sequence of objects. Definition 2. A List is either: (1) the empty list or (2) and object followed by another list ′. Are these equivalent definitions? 

Operators corresponding to Base Case and Constructor Case null: prepend ∶ × ⟶ 

More Operations over Lists null: prepend: × ⟶ first: List ⟶ rest: ⟶ empty: ⟶ 

Defining them Using Base and Constructor empty null ⟶ empty prepend (, ) ⟶ first prepend (, ) ⟶ rest prepend (, ) ⟶ 

Another operation: length null: prepend: × ⟶ first: List ⟶ rest: ⟶ empty: ⟶ length: ⟶ 

How to define length recursively? null: prepend: × ⟶ length: ⟶ 

Length of a List Definition. The length of a list, , is: 0 : if is null 1 + length : if = prepend , 10 

Another operation: Concatenation null: prepend: × ⟶ Concatenation: × ⟶ 

Concatenation: Descriptive Def Definition. The concatenation of two lists, = 1 , 2 , … , and = 1 , 2 , … , is 1 , 2 , … , , 1 , 2 , … , . How can we define this constructively? 12 

Concatenation Definition. The concatenation of two lists, = 1 , 2 , … , and = 1 , 2 , … , is 1 , 2 , … , , 1 , 2 , … , . How can we define this constructively? Also: Any questions about definitions so far, recursive data, etc. 13 

Concatenation: Descriptive Def Definition. The concatenation of two lists, = 1 , 2 , … , and = 1 , 2 , … , is 1 , 2 , … , , 1 , 2 , … , . How can we define this constructively? 14 Suggestion: Define p+q ::= Prepend(p,q) Why does not work: then first(p+q) would be p not 1 Another reason this is not a good idea to define concatenation: We might define Lists of “restricted objects” (e.g. integers) in which case Prepend(e,q) only accepts “e” to be in restricted form (not including lists) 

Concatenation: Constructive Def Definition. The concatenation ( + ) of two lists, and , is recursively defined as: Base case: if = (empty list) then + = Constructor case: if = prepend(, ) for some list , obj then + = prepend(, + ) 15 

Did we define length and concatenation properly? Prove. For any two lists, and , length( + ) = length() + length() 16 

Structural Induction To prove for all List 1. Prove for all base list . 2. Prove ⇒ for all constructable from . 17 

Prove. For any two lists, and , length( + ) = length() + length() Base case: = (empty list) + = Constructor case: = prepend (, ) + = prepend (, + ) 18 Proposition F(p) for any list p : F(p) : ∀ ∈ . length( + ) = length() + length() 

Structural Induction (Data Types) Invariant Principle (State Machines) (Regular) Induction (Natural Numbers) for all data type objects for all reachable states To prove ∙ for all natural numbers prove a base case 0 0 base object prove an inductive step ⇒ ( + 1) ⇒ for all constructable from ⇒ for all reachable from 19 

Charge Next week: Infinite sets and cardinalities.. Problem Set 6 due tomorrow (make sure to read the updated problem 7) Problem set 7 posted tomorrow night (or Sat morning) – read chapter 7 + notes/slides of this weak