### A Formal Proof of the Banach-Tarski Theorem in ACL2(r)

##### Reference: Weston, T.. “THE BANACH-TARSKI PARADOX.” (2003)


groups.lisp: A free group of reduced words.

free-group.lisp: A free group of 3-d matrices of rank 2.

rotations.lisp: 3-d rotation and its properties. Containts proof for the every element
		of the free group is a 3-d rotation.

hausdorff-paradox-1.lisp: Contains the proof of the equivalence between different partitions of S^2 minus the set of poles.

hausdorff-paradox-2.lisp: Proof for the set of poles is countable.

countable-sets.lisp: Contains proof for the cartesain product of two countable sets is countable.

banach-tarski-s2.lisp: Proof of the Banach-Tarski theorem on S^2.

banach-tarski-b-0.lisp: Proof of the Banach-Tarski theorem on the unit ball except for the origin.

banach-tarski.lisp: Proof of the Banach-Tarski theorem on the unit ball.
