User blog comment:Upquark11111/An Explanation of Loader's Number/@comment-11227630-20171210002438/@comment-11227630-20171210074355

I still didn't understand how does the "Iter" go wrong in λ→.

First, λ→ has at least 2 versions (3-rule version and λ-cube version) of inference ruleset.

The 3-rule ruleset is: while the λ-cube ruleset is: (where $$s\in\{*,\Box\}$$) Then there are 2 questions:
 * 1) (varible rule)$$\dfrac{(x:A)\in\Gamma}{\Gamma\vdash x:A}$$
 * 2) (application rule)$$\dfrac{\Gamma\vdash M:(A\rightarrow B)\quad\Gamma\vdash N:A}{\Gamma\vdash (MN):B}$$
 * 3) (lambda-rule)$$\dfrac{\Gamma,x:A\vdash M:B}{\Gamma\vdash(\lambda x:A.M):(A\rightarrow B)}$$
 * 1) (axiom)$$\vdash*:\Box$$
 * 2) (varible rule)$$\dfrac{\Gamma\vdash A:s}{\Gamma,x:A\vdash x:A}$$ where $$x\notin\Gamma$$
 * 3) (weakening)$$\dfrac{\Gamma\vdash M:A\quad\Gamma\vdash B:s}{\Gamma,x:B\vdash M:A}$$ where $$x\notin\Gamma$$
 * 4) (conversion)$$\dfrac{\Gamma\vdash M:A\quad\Gamma\vdash B:s\quad A=_\beta B}{\Gamma\vdash M:B}$$
 * 5) (application rule)$$\dfrac{\Gamma\vdash M:(\Pi x:A.B)\quad\Gamma\vdash N:A}{\Gamma\vdash(MN):B[x:=N]}$$
 * 6) (lambda rule)$$\dfrac{\Gamma\vdash(\Pi x:A.B):s\quad\Gamma,x:A\vdash M:B}{\Gamma\vdash(\lambda x:A.M):(\Pi x:A.B)}$$
 * 7) (pi rule)$$\dfrac{\Gamma\vdash A:*\quad\Gamma,x:A\vdash B:*}{\Gamma\vdash(\Pi x:A.B):*}$$
 * 1) Does the λ-cube version of λ→ have the same (i.e. tetrational) growth rate limit as the 3-rule version?
 * 2) What's the problem when infer the type of Iter in the λ-cube version of λ→?