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For the meridian case $(b : 2) \rightarrow \mathsf{north} \equiv \mathsf{south}$, we can do $2$-induction on $b$. First, for the $\mathsf{true}$ case, let's see the diagram of what we're trying to prove. +![](./goal3.jpg) + +After simplifying all the expressions, we find that $g(f(\mathsf{merid} \; \mathsf{true} \; i)) = g(\mathsf{base}) = \mathsf{north}$. +So really, we're trying to prove that $\mathsf{north} \equiv \mathsf{merid} \; \mathsf{true} \; i$. + +![](./goal4.jpg) + +But wait, that's odd... there's a $\mathsf{south}$ in the top right corner. What can we do? + +Well, we could actually use the _same_ path on the right as on the top. +We won't even need an $\mathsf{hcomp}$ for this, just a clever interval expression: + ``` gf (merid true i) j = merid true (i ∧ j) ``` -For the last part, we are trying to prove: +One last cube for the $g(f(\mathsf{merid} \; \mathsf{false} \; i)) \; j$ case. +This cube actually looks very similar to the $f(g(s))$ cube, although the points are in the suspension type rather than in $S^1$. +Once again we have a composition, so we will need an extra $\mathsf{hfill}$ to get us the front face of this cube. -`(merid false ∙ (λ i₁ → merid true (~ i₁))) i ≡ merid false i` +![](./gf.jpg) + +This cube cleanly extracts into the following Agda code. ``` gf (merid false i) j = @@ -280,3 +296,8 @@ For the last part, we are trying to prove: (j = i1) → merid false i in hcomp u (merid false i) ``` + +These are some of my takeaways from studying cubical type theory and $\mathsf{hcomp}$ these past few weeks. +One thing I'd have to note is that drawing all these cubes really does make me wish there was some kind of 3D visualizer for these cubes... + +Anyway, that's all! See you next post :)