  In this project, a rather complete proof-theoretical formalization of Lambek
Calculus (non-associative with arbitrary extensions) has been ported from Coq
proof assistent to HOL4 theorem prover, with some improvements and new
theorems.
  Three deduction systems (Syntactic Calculus, Natural Deduction and Sequent
Calculus) of Lambek Calculus are defined with many related theorems proved. The
equivalance between these systems are formally proved. Finally, a formalization
of Sequent Calculus proofs (where Coq has built-in supports) has been designed
and implemented in HOL4. Some basic results including the sub-formula
properties of the so-called "cut-free" proofs are formally proved.
  This work can be considered as the preliminary work towards a language parser
based on category grammars which is not multimodal but still has ability to
support context-sensitive languages through customized extensions.
