diff --git a/Library/Homebrew/ast_constants.rb b/Library/Homebrew/ast_constants.rb
index cc778677fa..9093aeab2b 100644
--- a/Library/Homebrew/ast_constants.rb
+++ b/Library/Homebrew/ast_constants.rb
@@ -1,9 +1,9 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "macos_version"
 
-FORMULA_COMPONENT_PRECEDENCE_LIST = [
+FORMULA_COMPONENT_PRECEDENCE_LIST = T.let([
   [{ name: :include,   type: :method_call }],
   [{ name: :desc,      type: :method_call }],
   [{ name: :homepage,  type: :method_call }],
@@ -49,4 +49,4 @@ FORMULA_COMPONENT_PRECEDENCE_LIST = [
   [{ name: :caveats, type: :method_definition }],
   [{ name: :plist_options, type: :method_call }, { name: :plist, type: :method_definition }],
   [{ name: :test, type: :block_call }],
-].freeze
+].freeze, T::Array[[{ name: Symbol, type: Symbol }]])
diff --git a/Library/Homebrew/cask/cache.rb b/Library/Homebrew/cask/cache.rb
index 62e7662e7f..d515d9d7a4 100644
--- a/Library/Homebrew/cask/cache.rb
+++ b/Library/Homebrew/cask/cache.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Cask
@@ -8,7 +8,7 @@ module Cask
   module Cache
     sig { returns(Pathname) }
     def self.path
-      @path ||= HOMEBREW_CACHE/"Cask"
+      @path ||= T.let(HOMEBREW_CACHE/"Cask", T.nilable(Pathname))
     end
   end
 end
diff --git a/Library/Homebrew/cmd/--cache.rb b/Library/Homebrew/cmd/--cache.rb
index d6a44f503a..944bb72da2 100644
--- a/Library/Homebrew/cmd/--cache.rb
+++ b/Library/Homebrew/cmd/--cache.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "fetch"
@@ -59,7 +59,7 @@ module Homebrew
     formulae_or_casks.each do |formula_or_cask|
       case formula_or_cask
       when Formula
-        formula = T.cast(formula_or_cask, Formula)
+        formula = formula_or_cask
         ref = formula.loaded_from_api? ? formula.full_name : formula.path
 
         os_arch_combinations.each do |os, arch|
diff --git a/Library/Homebrew/cmd/--repository.rb b/Library/Homebrew/cmd/--repository.rb
index 0d15e532ec..ead618e8f0 100644
--- a/Library/Homebrew/cmd/--repository.rb
+++ b/Library/Homebrew/cmd/--repository.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -19,6 +19,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def __repository
     args = __repository_args.parse
 
diff --git a/Library/Homebrew/cmd/analytics.rb b/Library/Homebrew/cmd/analytics.rb
index 7703cc2c5d..f5ac8ef8a0 100644
--- a/Library/Homebrew/cmd/analytics.rb
+++ b/Library/Homebrew/cmd/analytics.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -24,6 +24,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def analytics
     args = analytics_args.parse
 
diff --git a/Library/Homebrew/cmd/autoremove.rb b/Library/Homebrew/cmd/autoremove.rb
index 94dea99605..6f2f724bc1 100644
--- a/Library/Homebrew/cmd/autoremove.rb
+++ b/Library/Homebrew/cmd/autoremove.rb
@@ -1,13 +1,12 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cleanup"
 require "cli/parser"
 
 module Homebrew
-  module_function
-
-  def autoremove_args
+  sig { returns(CLI::Parser) }
+  def self.autoremove_args
     Homebrew::CLI::Parser.new do
       description <<~EOS
         Uninstall formulae that were only installed as a dependency of another formula and are now no longer needed.
@@ -19,7 +18,8 @@ module Homebrew
     end
   end
 
-  def autoremove
+  sig { void }
+  def self.autoremove
     args = autoremove_args.parse
 
     Cleanup.autoremove(dry_run: args.dry_run?)
diff --git a/Library/Homebrew/cmd/cleanup.rb b/Library/Homebrew/cmd/cleanup.rb
index ebc80f8607..225696d51d 100644
--- a/Library/Homebrew/cmd/cleanup.rb
+++ b/Library/Homebrew/cmd/cleanup.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cleanup"
@@ -33,6 +33,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def cleanup
     args = cleanup_args.parse
 
diff --git a/Library/Homebrew/cmd/commands.rb b/Library/Homebrew/cmd/commands.rb
index 5f87e44284..f28546ade6 100644
--- a/Library/Homebrew/cmd/commands.rb
+++ b/Library/Homebrew/cmd/commands.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -22,6 +22,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def commands
     args = commands_args.parse
 
diff --git a/Library/Homebrew/cmd/completions.rb b/Library/Homebrew/cmd/completions.rb
index c9655658e0..ac7a8e2a5f 100644
--- a/Library/Homebrew/cmd/completions.rb
+++ b/Library/Homebrew/cmd/completions.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -25,6 +25,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def completions
     args = completions_args.parse
 
diff --git a/Library/Homebrew/cmd/config.rb b/Library/Homebrew/cmd/config.rb
index b71022140c..d75bc5ce54 100644
--- a/Library/Homebrew/cmd/config.rb
+++ b/Library/Homebrew/cmd/config.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "system_config"
@@ -19,6 +19,7 @@ module Homebrew
     end
   end
 
+  sig { void }
   def config
     config_args.parse
 
diff --git a/Library/Homebrew/cmd/docs.rb b/Library/Homebrew/cmd/docs.rb
index f3d7d713d1..b8107aea66 100644
--- a/Library/Homebrew/cmd/docs.rb
+++ b/Library/Homebrew/cmd/docs.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
diff --git a/Library/Homebrew/cmd/help.rb b/Library/Homebrew/cmd/help.rb
index bcad2600c1..2d286ad465 100644
--- a/Library/Homebrew/cmd/help.rb
+++ b/Library/Homebrew/cmd/help.rb
@@ -1,9 +1,10 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "help"
 
 module Homebrew
+  sig { returns(T.noreturn) }
   def help
     Help.help
   end
diff --git a/Library/Homebrew/default_prefix.rb b/Library/Homebrew/default_prefix.rb
index 6aaf1351a5..91c9f878d6 100644
--- a/Library/Homebrew/default_prefix.rb
+++ b/Library/Homebrew/default_prefix.rb
@@ -1,7 +1,7 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
-  DEFAULT_PREFIX = ENV.fetch("HOMEBREW_DEFAULT_PREFIX").freeze
-  DEFAULT_REPOSITORY = ENV.fetch("HOMEBREW_DEFAULT_REPOSITORY").freeze
+  DEFAULT_PREFIX = T.let(ENV.fetch("HOMEBREW_DEFAULT_PREFIX").freeze, String)
+  DEFAULT_REPOSITORY = T.let(ENV.fetch("HOMEBREW_DEFAULT_REPOSITORY").freeze, String)
 end
diff --git a/Library/Homebrew/extend/cachable.rb b/Library/Homebrew/extend/cachable.rb
index c90e9ca474..b124c638e6 100644
--- a/Library/Homebrew/extend/cachable.rb
+++ b/Library/Homebrew/extend/cachable.rb
@@ -1,11 +1,13 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Cachable
+  sig { returns(T::Hash[T.untyped, T.untyped]) }
   def cache
-    @cache ||= {}
+    @cache ||= T.let({}, T.nilable(T::Hash[T.untyped, T.untyped]))
   end
 
+  sig { void }
   def clear_cache
     cache.clear
   end
diff --git a/Library/Homebrew/extend/os/cleanup.rb b/Library/Homebrew/extend/os/cleanup.rb
index 983b23a2ea..2018cbb911 100644
--- a/Library/Homebrew/extend/os/cleanup.rb
+++ b/Library/Homebrew/extend/os/cleanup.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 if OS.mac?
diff --git a/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb b/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
index 4072a84b44..792e3458a1 100644
--- a/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
+++ b/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
@@ -1,10 +1,11 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
   class << self
     alias generic_git_tags git_tags
 
+    sig { returns(String) }
     def git_tags
       tags = generic_git_tags
       tags = Utils.popen_read("git tag --list | sort -rV") if tags.blank?
diff --git a/Library/Homebrew/extend/os/parser.rb b/Library/Homebrew/extend/os/parser.rb
index 97225de0c8..eb84af9d0a 100644
--- a/Library/Homebrew/extend/os/parser.rb
+++ b/Library/Homebrew/extend/os/parser.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "extend/os/linux/parser" if OS.linux?
diff --git a/Library/Homebrew/extend/predicable.rb b/Library/Homebrew/extend/predicable.rb
index 6aa79ad949..7eb7c9edbf 100644
--- a/Library/Homebrew/extend/predicable.rb
+++ b/Library/Homebrew/extend/predicable.rb
@@ -1,7 +1,8 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Predicable
+  sig { params(attrs: Symbol).void }
   def attr_predicate(*attrs)
     attrs.each do |attr|
       define_method attr do
diff --git a/Library/Homebrew/formula_free_port.rb b/Library/Homebrew/formula_free_port.rb
index 91e94da609..8659c74c5c 100644
--- a/Library/Homebrew/formula_free_port.rb
+++ b/Library/Homebrew/formula_free_port.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "socket"
@@ -10,6 +10,7 @@ module Homebrew
   module FreePort
     # Returns a free port.
     # @api public
+    sig { returns(Integer) }
     def free_port
       server = TCPServer.new 0
       _, port, = server.addr
diff --git a/Library/Homebrew/livecheck/constants.rb b/Library/Homebrew/livecheck/constants.rb
index f203ab67b5..09645c4229 100644
--- a/Library/Homebrew/livecheck/constants.rb
+++ b/Library/Homebrew/livecheck/constants.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
diff --git a/Library/Homebrew/source_location.rb b/Library/Homebrew/source_location.rb
index 6b7bc7e5f6..9b4b4b7789 100644
--- a/Library/Homebrew/source_location.rb
+++ b/Library/Homebrew/source_location.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
diff --git a/Library/Homebrew/tap_constants.rb b/Library/Homebrew/tap_constants.rb
index b5195f3949..f7d4e7c90f 100644
--- a/Library/Homebrew/tap_constants.rb
+++ b/Library/Homebrew/tap_constants.rb
@@ -1,17 +1,20 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 # Match taps' formulae, e.g. `someuser/sometap/someformula`
-HOMEBREW_TAP_FORMULA_REGEX = %r{^([\w-]+)/([\w-]+)/([\w+-.@]+)$}.freeze
+HOMEBREW_TAP_FORMULA_REGEX = T.let(%r{^([\w-]+)/([\w-]+)/([\w+-.@]+)$}.freeze, Regexp)
 # Match taps' casks, e.g. `someuser/sometap/somecask`
-HOMEBREW_TAP_CASK_REGEX = %r{^([\w-]+)/([\w-]+)/([a-z0-9\-_]+)$}.freeze
+HOMEBREW_TAP_CASK_REGEX = T.let(%r{^([\w-]+)/([\w-]+)/([a-z0-9\-_]+)$}.freeze, Regexp)
 # Match main cask taps' casks, e.g. `homebrew/cask/somecask` or `somecask`
-HOMEBREW_MAIN_TAP_CASK_REGEX = %r{^([Hh]omebrew/(?:homebrew-)?cask/)?[a-z0-9\-_]+$}.freeze
+HOMEBREW_MAIN_TAP_CASK_REGEX = T.let(%r{^([Hh]omebrew/(?:homebrew-)?cask/)?[a-z0-9\-_]+$}.freeze, Regexp)
 # Match taps' directory paths, e.g. `HOMEBREW_LIBRARY/Taps/someuser/sometap`
-HOMEBREW_TAP_DIR_REGEX = %r{#{Regexp.escape(HOMEBREW_LIBRARY.to_s)}/Taps/(?<user>[\w-]+)/(?<repo>[\w-]+)}.freeze
+HOMEBREW_TAP_DIR_REGEX = T.let(
+  %r{#{Regexp.escape(HOMEBREW_LIBRARY.to_s)}/Taps/(?<user>[\w-]+)/(?<repo>[\w-]+)}.freeze, Regexp
+)
 # Match taps' formula paths, e.g. `HOMEBREW_LIBRARY/Taps/someuser/sometap/someformula`
-HOMEBREW_TAP_PATH_REGEX = Regexp.new(HOMEBREW_TAP_DIR_REGEX.source + %r{(?:/.*)?$}.source).freeze
+HOMEBREW_TAP_PATH_REGEX = T.let(Regexp.new(HOMEBREW_TAP_DIR_REGEX.source + %r{(?:/.*)?$}.source).freeze, Regexp)
 # Match official taps' casks, e.g. `homebrew/cask/somecask or homebrew/cask-versions/somecask`
 HOMEBREW_CASK_TAP_CASK_REGEX =
-  %r{^(?:([Cc]askroom)/(cask|versions)|([Hh]omebrew)/(?:homebrew-)?(cask|cask-[\w-]+))/([\w+-.]+)$}.freeze
-HOMEBREW_OFFICIAL_REPO_PREFIXES_REGEX = /^(home|linux)brew-/.freeze
+  T.let(%r{^(?:([Cc]askroom)/(cask|versions)|([Hh]omebrew)/(?:homebrew-)?(cask|cask-[\w-]+))/([\w+-.]+)$}.freeze,
+        Regexp)
+HOMEBREW_OFFICIAL_REPO_PREFIXES_REGEX = T.let(/^(home|linux)brew-/.freeze, Regexp)
diff --git a/Library/Homebrew/test/support/lib/default_prefix.rb b/Library/Homebrew/test/support/lib/default_prefix.rb
index e912bd4831..9da2beeb83 100644
--- a/Library/Homebrew/test/support/lib/default_prefix.rb
+++ b/Library/Homebrew/test/support/lib/default_prefix.rb
@@ -1,8 +1,8 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
   # For testing's sake always assume the default prefix
-  DEFAULT_PREFIX = HOMEBREW_PREFIX.to_s.freeze
-  DEFAULT_REPOSITORY = HOMEBREW_REPOSITORY.to_s.freeze
+  DEFAULT_PREFIX = T.let(HOMEBREW_PREFIX.to_s.freeze, String)
+  DEFAULT_REPOSITORY = T.let(HOMEBREW_REPOSITORY.to_s.freeze, String)
 end
diff --git a/Library/Homebrew/test/utils/repology_spec.rb b/Library/Homebrew/test/utils/repology_spec.rb
index 0903f3a76a..97c22e8864 100644
--- a/Library/Homebrew/test/utils/repology_spec.rb
+++ b/Library/Homebrew/test/utils/repology_spec.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "utils/repology"
diff --git a/Library/Homebrew/utils/rubocop.rb b/Library/Homebrew/utils/rubocop.rb
index fe98ae580d..4e760a601e 100755
--- a/Library/Homebrew/utils/rubocop.rb
+++ b/Library/Homebrew/utils/rubocop.rb
@@ -1,5 +1,5 @@
 #!/usr/bin/env ruby
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require_relative "../standalone"