add mdbook compilation check to CI (#45)

Reasoning: till now we're only compiling it when the changes land at the
`main` branch, which allows invalid changes to the mdbook to pass the PR
check. With the update at this commit we ensure that the new PRs pass
the mdbook compilation check.

.github/workflows/mdbook-publish.yml

@@ -0,0 +1,56 @@
+name: Publish mdbook
+on:
+  push:
+    branches: ["main"]
+
+  # Allows to run this workflow manually from the Actions tab
+  workflow_dispatch:
+
+# Sets permissions of the GITHUB_TOKEN to allow deployment to GitHub Pages
+permissions:
+  contents: read
+  pages: write
+  id-token: write
+
+# Allow only one concurrent deployment, skipping runs queued between the run in-progress and latest queued.
+# However, do NOT cancel in-progress runs as we want to allow these production deployments to complete.
+concurrency:
+  group: "pages"
+  cancel-in-progress: false
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+    env:
+      MDBOOK_VERSION: 0.4.40
+      MDBOOKKATEX_VERSION: 0.7.0
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install mdBook
+        run: |
+          curl --proto '=https' --tlsv1.2 https://sh.rustup.rs -sSf -y | sh
+          rustup update
+          cargo install --version ${MDBOOK_VERSION} mdbook
+          cargo install --version ${MDBOOKKATEX_VERSION} mdbook-katex
+      - name: Setup Pages
+        id: pages
+        uses: actions/configure-pages@v4
+      - name: Build with mdBook
+        run: |
+          cd book
+          mdbook build
+      - name: Upload artifact
+        uses: actions/upload-pages-artifact@v3
+        with:
+          path: ./book/book
+
+  deploy:
+    environment:
+      name: github-pages
+      url: ${{ steps.deployment.outputs.page_url }}
+    runs-on: ubuntu-latest
+    needs: build
+    steps:
+      - name: Deploy to GitHub Pages
+        id: deployment
+        uses: actions/deploy-pages@v4