Finite promise games

The game G(T,n,s) with n,s ≥ 1 is a game with n alternating plays by Alice and Bob. At every stage of the game, Alice can offer Bob a number x in the range [0,s] of the form y+z or w! with y and z numbers played earlier in the game by Bob. If Bob accept x, he promises that there will be no T inversion of x, and he plays x. If Bob reject x, he plays a T inversion of x and promises that x is never played by Bob. Bob wins the game if and only if he kept all his promises.

A T inversion of x is some linear function with arguments smaller than x, defined with the set T which is defined before the game. The numbers from this extremely fast growing computable function may exceed the growth rate of Loader.c.