diff --git a/Library/Homebrew/cask/dsl/conflicts_with.rb b/Library/Homebrew/cask/dsl/conflicts_with.rb
index da47113ab5..99ca4c9838 100644
--- a/Library/Homebrew/cask/dsl/conflicts_with.rb
+++ b/Library/Homebrew/cask/dsl/conflicts_with.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "delegate"
@@ -24,14 +24,14 @@ module Cask
       def initialize(**options)
         options.assert_valid_keys!(*VALID_KEYS)
 
-        conflicts = options.transform_values { |v| Set.new(Array(v)) }
+        conflicts = options.transform_values { |v| Set.new(Kernel.Array(v)) }
         conflicts.default = Set.new
 
         super(conflicts)
       end
 
       def to_json(generator)
-        transform_values(&:to_a).to_json(generator)
+        __getobj__.transform_values(&:to_a).to_json(generator)
       end
     end
   end
diff --git a/Library/Homebrew/cask/dsl/depends_on.rb b/Library/Homebrew/cask/dsl/depends_on.rb
index 48c96b893f..9522f0553e 100644
--- a/Library/Homebrew/cask/dsl/depends_on.rb
+++ b/Library/Homebrew/cask/dsl/depends_on.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "delegate"
@@ -38,7 +38,7 @@ module Cask
         pairs.each do |key, value|
           raise "invalid depends_on key: '#{key.inspect}'" unless VALID_KEYS.include?(key)
 
-          self[key] = send(:"#{key}=", *value)
+          __getobj__[key] = send(:"#{key}=", *value)
         end
       end
 
@@ -54,15 +54,18 @@ module Cask
       def macos=(*args)
         raise "Only a single 'depends_on macos' is allowed." if defined?(@macos)
 
+        # workaround for https://github.com/sorbet/sorbet/issues/6860
+        first_arg = args.first&.to_s
+
         begin
           @macos = if args.count > 1
             MacOSRequirement.new([args], comparator: "==")
           elsif MacOSVersions::SYMBOLS.key?(args.first)
             MacOSRequirement.new([args.first], comparator: "==")
-          elsif /^\s*(?<comparator><|>|[=<>]=)\s*:(?<version>\S+)\s*$/ =~ args.first
-            MacOSRequirement.new([version.to_sym], comparator: comparator)
-          elsif /^\s*(?<comparator><|>|[=<>]=)\s*(?<version>\S+)\s*$/ =~ args.first
-            MacOSRequirement.new([version], comparator: comparator)
+          elsif (md = /^\s*(?<comparator><|>|[=<>]=)\s*:(?<version>\S+)\s*$/.match(first_arg))
+            MacOSRequirement.new([T.must(md[:version]).to_sym], comparator: md[:comparator])
+          elsif (md = /^\s*(?<comparator><|>|[=<>]=)\s*(?<version>\S+)\s*$/.match(first_arg))
+            MacOSRequirement.new([md[:version]], comparator: md[:comparator])
           else # rubocop:disable Lint/DuplicateBranch
             MacOSRequirement.new([args.first], comparator: "==")
           end
diff --git a/Library/Homebrew/cask/dsl/depends_on.rbi b/Library/Homebrew/cask/dsl/depends_on.rbi
new file mode 100644
index 0000000000..92d6827c64
--- /dev/null
+++ b/Library/Homebrew/cask/dsl/depends_on.rbi
@@ -0,0 +1,5 @@
+# typed: strict
+
+class Cask::DSL::DependsOn
+  include Kernel
+end
diff --git a/Library/Homebrew/cask/installer.rb b/Library/Homebrew/cask/installer.rb
index 8f9f756912..1a231504d5 100644
--- a/Library/Homebrew/cask/installer.rb
+++ b/Library/Homebrew/cask/installer.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "formula_installer"
@@ -241,7 +241,7 @@ module Cask
       save_download_sha if @cask.version.latest?
     rescue => e
       begin
-        already_installed_artifacts.each do |artifact|
+        already_installed_artifacts&.each do |artifact|
           if artifact.respond_to?(:uninstall_phase)
             odebug "Reverting installation of artifact of class #{artifact.class}"
             artifact.uninstall_phase(command: @command, verbose: verbose?, force: force?)
@@ -296,7 +296,7 @@ module Cask
 
       graph = ::Utils::TopologicalHash.graph_package_dependencies(@cask)
 
-      raise CaskSelfReferencingDependencyError, cask.token if graph[@cask].include?(@cask)
+      raise CaskSelfReferencingDependencyError, @cask.token if graph[@cask].include?(@cask)
 
       ::Utils::TopologicalHash.graph_package_dependencies(primary_container.dependencies, graph)
 
diff --git a/Library/Homebrew/cask/quarantine.rb b/Library/Homebrew/cask/quarantine.rb
index 2b9ec87058..7333b4e231 100644
--- a/Library/Homebrew/cask/quarantine.rb
+++ b/Library/Homebrew/cask/quarantine.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "development_tools"
@@ -11,29 +11,27 @@ module Cask
   module Quarantine
     extend T::Sig
 
-    module_function
-
     QUARANTINE_ATTRIBUTE = "com.apple.quarantine"
 
     QUARANTINE_SCRIPT = (HOMEBREW_LIBRARY_PATH/"cask/utils/quarantine.swift").freeze
 
-    def swift
+    def self.swift
       @swift ||= DevelopmentTools.locate("swift")
     end
-    private :swift
+    private_class_method :swift
 
-    def xattr
+    def self.xattr
       @xattr ||= DevelopmentTools.locate("xattr")
     end
-    private :xattr
+    private_class_method :xattr
 
-    def swift_target_args
+    def self.swift_target_args
       ["-target", "#{Hardware::CPU.arch}-apple-macosx#{MacOS.version}"]
     end
-    private :swift_target_args
+    private_class_method :swift_target_args
 
     sig { returns(Symbol) }
-    def check_quarantine_support
+    def self.check_quarantine_support
       odebug "Checking quarantine support"
 
       if !system_command(xattr, args: ["-h"], print_stderr: false).success?
@@ -58,13 +56,13 @@ module Cask
       end
     end
 
-    def available?
+    def self.available?
       @status ||= check_quarantine_support
 
       @status == :quarantine_available
     end
 
-    def detect(file)
+    def self.detect(file)
       return if file.nil?
 
       odebug "Verifying Gatekeeper status of #{file}"
@@ -76,13 +74,13 @@ module Cask
       quarantine_status
     end
 
-    def status(file)
+    def self.status(file)
       system_command(xattr,
                      args:         ["-p", QUARANTINE_ATTRIBUTE, file],
                      print_stderr: false).stdout.rstrip
     end
 
-    def toggle_no_translocation_bit(attribute)
+    def self.toggle_no_translocation_bit(attribute)
       fields = attribute.split(";")
 
       # Fields: status, epoch, download agent, event ID
@@ -94,7 +92,7 @@ module Cask
       fields.join(";")
     end
 
-    def release!(download_path: nil)
+    def self.release!(download_path: nil)
       return unless detect(download_path)
 
       odebug "Releasing #{download_path} from quarantine"
@@ -112,7 +110,7 @@ module Cask
       raise CaskQuarantineReleaseError.new(download_path, quarantiner.stderr)
     end
 
-    def cask!(cask: nil, download_path: nil, action: true)
+    def self.cask!(cask: nil, download_path: nil, action: true)
       return if cask.nil? || download_path.nil?
 
       return if detect(download_path)
@@ -139,7 +137,7 @@ module Cask
       end
     end
 
-    def propagate(from: nil, to: nil)
+    def self.propagate(from: nil, to: nil)
       return if from.nil? || to.nil?
 
       raise CaskError, "#{from} was not quarantined properly." unless detect(from)
diff --git a/Library/Homebrew/cask/upgrade.rb b/Library/Homebrew/cask/upgrade.rb
index a27925b921..17c38f43be 100644
--- a/Library/Homebrew/cask/upgrade.rb
+++ b/Library/Homebrew/cask/upgrade.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "env_config"
@@ -124,7 +124,9 @@ module Cask
 
       return true if caught_exceptions.empty?
       raise MultipleCaskErrors, caught_exceptions if caught_exceptions.count > 1
-      raise caught_exceptions.first if caught_exceptions.count == 1
+      raise caught_exceptions.fetch(0) if caught_exceptions.count == 1
+
+      false
     end
 
     def self.upgrade_cask(
diff --git a/Library/Homebrew/extend/git_repository.rbi b/Library/Homebrew/extend/git_repository.rbi
index 534b936b96..9ffee241cf 100644
--- a/Library/Homebrew/extend/git_repository.rbi
+++ b/Library/Homebrew/extend/git_repository.rbi
@@ -1,8 +1,5 @@
 # typed: strict
 
 module GitRepositoryExtension
-  include Kernel
-
-  sig { params(args: T.any(String, Pathname)).returns(Pathname) }
-  def join(*args); end
+  requires_ancestor { Pathname }
 end
diff --git a/Library/Homebrew/extend/pathname.rbi b/Library/Homebrew/extend/pathname.rbi
index 12a94cd3eb..a682b461ee 100644
--- a/Library/Homebrew/extend/pathname.rbi
+++ b/Library/Homebrew/extend/pathname.rbi
@@ -1,19 +1,9 @@
 # typed: strict
 
 module DiskUsageExtension
-  include Kernel
-
-  def exist?; end
-
-  def symlink?; end
-
-  def resolved_path; end
+  requires_ancestor { Pathname }
 end
 
 module ObserverPathnameExtension
-  include Kernel
-
-  def dirname; end
-
-  def basename; end
+  requires_ancestor { Pathname }
 end
diff --git a/Library/Homebrew/os/mac/version.rb b/Library/Homebrew/os/mac/version.rb
index fd709f72f2..d407e882ac 100644
--- a/Library/Homebrew/os/mac/version.rb
+++ b/Library/Homebrew/os/mac/version.rb
@@ -45,7 +45,7 @@ module OS
       sig { returns(T.self_type) }
       def strip_patch
         # Big Sur is 11.x but Catalina is 10.15.x.
-        if major >= 11
+        if T.must(major) >= 11
           self.class.new(major.to_s)
         else
           major_minor
diff --git a/Library/Homebrew/os/mac/version.rbi b/Library/Homebrew/os/mac/version.rbi
deleted file mode 100644
index dc7c90848a..0000000000
--- a/Library/Homebrew/os/mac/version.rbi
+++ /dev/null
@@ -1,10 +0,0 @@
-# typed: strict
-
-module OS
-  module Mac
-    class Version
-      sig { returns(Token) }
-      def major; end
-    end
-  end
-end