Merge

General Information

Algorithmic problem: Merging two sorted sequences

Type of algorithm: loop

Auxiliary data: There are two current positions: [math]\displaystyle{ i_1 \in \{0,\dots,|S_1|\} }[/math] and [math]\displaystyle{ i_2 \in \{0,\dots,|S_2|\} }[/math] .

Abstract View

Invariant: Before and after each iteration:

1. [math]\displaystyle{ S }[/math] consists exactly of all elements [math]\displaystyle{ S_1[1],\dots,S_1[i_1] }[/math] and [math]\displaystyle{ S_2[1],\dots,S_2[i_2] }[/math].
2. [math]\displaystyle{ S }[/math] is sorted according to the comparison on [math]\displaystyle{ S_1 }[/math] and [math]\displaystyle{ S_2 }[/math].

Variant: [math]\displaystyle{ i_1 + i_2 }[/math] increases by [math]\displaystyle{ 1 }[/math]; neither [math]\displaystyle{ i_1 }[/math] nor [math]\displaystyle{ i_2 }[/math] decreases.

Break condition: [math]\displaystyle{ i_1 = |S_1| }[/math] and [math]\displaystyle{ i_2 = |S_2| }[/math].

Induction Basis

Abstract view: [math]\displaystyle{ i_1 := 0 }[/math] and [math]\displaystyle{ i_2 := 0 }[/math].

Implementation: Obvious.

Proof: Nothing to show.