rustdoc: add nobuild typescript checking to our JS

src/ci/docker/host-x86_64/mingw-check/Dockerfile

@@ -29,7 +29,7 @@ ENV PATH="/node/bin:${PATH}"
 
 # Install es-check
 # Pin its version to prevent unrelated CI failures due to future es-check versions.
-RUN npm install es-check@6.1.1 eslint@8.6.0 -g
+RUN npm install es-check@6.1.1 eslint@8.6.0 typescript@5.7.3 -g
 
 COPY scripts/sccache.sh /scripts/
 RUN sh /scripts/sccache.sh
@@ -68,4 +68,5 @@ ENV SCRIPT \
            es-check es2019 ../src/librustdoc/html/static/js/*.js && \
            eslint -c ../src/librustdoc/html/static/.eslintrc.js ../src/librustdoc/html/static/js/*.js && \
            eslint -c ../src/tools/rustdoc-js/.eslintrc.js ../src/tools/rustdoc-js/tester.js && \
-           eslint -c ../src/tools/rustdoc-gui/.eslintrc.js ../src/tools/rustdoc-gui/tester.js
+           eslint -c ../src/tools/rustdoc-gui/.eslintrc.js ../src/tools/rustdoc-gui/tester.js && \
+           tsc --project ../src/librustdoc/html/static/js/tsconfig.json

src/librustdoc/html/static/js/README.md

@@ -3,13 +3,9 @@
 These JavaScript files are incorporated into the rustdoc binary at build time,
 and are minified and written to the filesystem as part of the doc build process.
 
-We use the [Closure Compiler](https://github.com/google/closure-compiler/wiki/Annotating-JavaScript-for-the-Closure-Compiler)
+We use the [TypeScript Compiler](https://www.typescriptlang.org/docs/handbook/jsdoc-supported-types.html)
 dialect of JSDoc to comment our code and annotate params and return types.
 To run a check:
 
-    ./x.py doc library/std
-    npm i -g google-closure-compiler
-    google-closure-compiler -W VERBOSE \
-      build/<YOUR PLATFORM>/doc/{search-index*.js,crates*.js} \
-      src/librustdoc/html/static/js/{search.js,main.js,storage.js} \
-      --externs src/librustdoc/html/static/js/externs.js >/dev/null
+    npm i -g typescript
+    tsc --project tsconfig.json