diff --git a/Library/.rubocop.yml b/Library/.rubocop.yml
index c9134c5611..fb376dda25 100644
--- a/Library/.rubocop.yml
+++ b/Library/.rubocop.yml
@@ -69,7 +69,7 @@ Naming/HeredocDelimiterNaming:
 
 Naming/MethodName:
   IgnoredPatterns:
-    - '\AHEAD\?\Z'
+    - '\A(fetch_)?HEAD\?\Z'
 
 # Allow dashes in filenames.
 Naming/FileName:
diff --git a/Library/Homebrew/cask/caskroom.rb b/Library/Homebrew/cask/caskroom.rb
index ab9c1b4dc0..4aa3ea067b 100644
--- a/Library/Homebrew/cask/caskroom.rb
+++ b/Library/Homebrew/cask/caskroom.rb
@@ -8,6 +8,8 @@ module Cask
   #
   # @api private
   module Caskroom
+    extend T::Sig
+
     module_function
 
     def path
@@ -30,6 +32,7 @@ module Cask
       SystemCommand.run("/usr/bin/chgrp", args: ["admin", path], sudo: sudo)
     end
 
+    sig { params(config: Config).returns(T::Array[Cask]) }
     def casks(config: nil)
       return [] unless path.exist?
 
diff --git a/Library/Homebrew/cask/cmd/upgrade.rb b/Library/Homebrew/cask/cmd/upgrade.rb
index 582d2c743e..28bc7b6a55 100644
--- a/Library/Homebrew/cask/cmd/upgrade.rb
+++ b/Library/Homebrew/cask/cmd/upgrade.rb
@@ -26,6 +26,7 @@ module Cask
         }],
       ].freeze
 
+      sig { returns(Homebrew::CLI::Parser) }
       def self.parser
         super do
           switch "--force",
@@ -56,6 +57,20 @@ module Cask
         )
       end
 
+      sig do
+        params(
+          casks:          Cask,
+          args:           Homebrew::CLI::Args,
+          force:          T.nilable(T::Boolean),
+          greedy:         T.nilable(T::Boolean),
+          dry_run:        T.nilable(T::Boolean),
+          skip_cask_deps: T.nilable(T::Boolean),
+          verbose:        T.nilable(T::Boolean),
+          binaries:       T.nilable(T::Boolean),
+          quarantine:     T.nilable(T::Boolean),
+          require_sha:    T.nilable(T::Boolean),
+        ).void
+      end
       def self.upgrade_casks(
         *casks,
         args:,
diff --git a/Library/Homebrew/cli/args.rb b/Library/Homebrew/cli/args.rb
index d8f69c6077..b12f32712b 100644
--- a/Library/Homebrew/cli/args.rb
+++ b/Library/Homebrew/cli/args.rb
@@ -24,7 +24,7 @@ module Homebrew
 
         # Can set these because they will be overwritten by freeze_named_args!
         # (whereas other values below will only be overwritten if passed).
-        self[:named_args] = NamedArgs.new(parent: self)
+        self[:named] = NamedArgs.new(parent: self)
         self[:remaining] = []
       end
 
@@ -33,7 +33,7 @@ module Homebrew
       end
 
       def freeze_named_args!(named_args)
-        self[:named_args] = NamedArgs.new(
+        self[:named] = NamedArgs.new(
           *named_args.freeze,
           override_spec: spec(nil),
           force_bottle:  force_bottle?,
@@ -53,8 +53,9 @@ module Homebrew
         @flags_only = cli_args.select { |a| a.start_with?("--") }.freeze
       end
 
+      sig { returns(NamedArgs) }
       def named
-        named_args
+        self[:named]
       end
 
       def no_named?
diff --git a/Library/Homebrew/cli/args.rbi b/Library/Homebrew/cli/args.rbi
index 2314e9c8f3..b072016115 100644
--- a/Library/Homebrew/cli/args.rbi
+++ b/Library/Homebrew/cli/args.rbi
@@ -3,27 +3,62 @@
 module Homebrew
   module CLI
     class Args < OpenStruct
+      sig { returns(T.nilable(T::Boolean)) }
       def devel?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def HEAD?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def include_test?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def build_bottle?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def build_universal?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def build_from_source?; end
 
-      def named_args; end
-
+      sig { returns(T.nilable(T::Boolean)) }
       def force_bottle?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def debug?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def quiet?; end
 
+      sig { returns(T.nilable(T::Boolean)) }
       def verbose?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def fetch_HEAD?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def cask?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def dry_run?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def skip_cask_deps?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def greedy?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def force?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def ignore_pinned?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def display_times?; end
+
+      sig { returns(T.nilable(T::Boolean)) }
+      def formula?; end
     end
   end
 end
diff --git a/Library/Homebrew/cli/named_args.rb b/Library/Homebrew/cli/named_args.rb
index 99ef08e341..c179ee8a6c 100644
--- a/Library/Homebrew/cli/named_args.rb
+++ b/Library/Homebrew/cli/named_args.rb
@@ -13,7 +13,7 @@ module Homebrew
     # Helper class for loading formulae/casks from named arguments.
     #
     # @api private
-    class NamedArgs < SimpleDelegator
+    class NamedArgs < Array
       def initialize(*args, parent: Args.new, override_spec: nil, force_bottle: false, flags: [])
         @args = args
         @override_spec = override_spec
diff --git a/Library/Homebrew/cmd/upgrade.rb b/Library/Homebrew/cmd/upgrade.rb
index ce8ec7bf8a..f7c8da355e 100644
--- a/Library/Homebrew/cmd/upgrade.rb
+++ b/Library/Homebrew/cmd/upgrade.rb
@@ -92,6 +92,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def upgrade
     args = upgrade_args.parse
 
@@ -109,6 +110,7 @@ module Homebrew
     upgrade_outdated_casks(casks, args: args) unless upgrade_formulae
   end
 
+  sig { params(formulae: T::Array[Formula], args: CLI::Args).void }
   def upgrade_outdated_formulae(formulae, args:)
     return if args.cask?
 
@@ -176,6 +178,7 @@ module Homebrew
     Homebrew.messages.display_messages(display_times: args.display_times?)
   end
 
+  sig { params(casks: T::Array[Cask::Cask], args: CLI::Args).void }
   def upgrade_outdated_casks(casks, args:)
     return if args.formula?
 
diff --git a/Library/Homebrew/test/cli/parser_spec.rb b/Library/Homebrew/test/cli/parser_spec.rb
index 7f1c71c3ad..e5fae5af8e 100644
--- a/Library/Homebrew/test/cli/parser_spec.rb
+++ b/Library/Homebrew/test/cli/parser_spec.rb
@@ -62,7 +62,7 @@ describe Homebrew::CLI::Parser do
       it "passes through invalid options" do
         args = parser.parse(["-v", "named-arg", "--not-a-valid-option"], ignore_invalid_options: true)
         expect(args.remaining).to eq ["named-arg", "--not-a-valid-option"]
-        expect(args.named_args).to be_empty
+        expect(args.named).to be_empty
       end
     end