diff --git a/Library/Homebrew/cmd/postgresql-upgrade-database.rb b/Library/Homebrew/cmd/postgresql-upgrade-database.rb
index 50421460d9..e0c2907c97 100755
--- a/Library/Homebrew/cmd/postgresql-upgrade-database.rb
+++ b/Library/Homebrew/cmd/postgresql-upgrade-database.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -60,10 +60,10 @@ module Homebrew
 
     odie "No #{name} #{pg_version_data}.* version installed!" unless old_bin
 
-    server_stopped = false
-    moved_data = false
-    initdb_run = false
-    upgraded = false
+    server_stopped = T.let(false, T::Boolean)
+    moved_data = T.let(false, T::Boolean)
+    initdb_run = T.let(false, T::Boolean)
+    upgraded = T.let(false, T::Boolean)
 
     begin
       # Following instructions from:
@@ -88,7 +88,7 @@ module Homebrew
         system "#{old_bin}/pg_ctl", "-w", "-D", datadir, "start"
       end
 
-      initdb_args = []
+      initdb_args = T.let([], T::Array[String])
       locale_settings = %w[
         lc_collate
         lc_ctype
diff --git a/Library/Homebrew/dependencies.rb b/Library/Homebrew/dependencies.rb
index 3b107e9999..fedb68cb72 100644
--- a/Library/Homebrew/dependencies.rb
+++ b/Library/Homebrew/dependencies.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "delegate"
@@ -16,19 +16,19 @@ class Dependencies < SimpleDelegator
   alias eql? ==
 
   def optional
-    select(&:optional?)
+    __getobj__.select(&:optional?)
   end
 
   def recommended
-    select(&:recommended?)
+    __getobj__.select(&:recommended?)
   end
 
   def build
-    select(&:build?)
+    __getobj__.select(&:build?)
   end
 
   def required
-    select(&:required?)
+    __getobj__.select(&:required?)
   end
 
   def default
@@ -37,7 +37,7 @@ class Dependencies < SimpleDelegator
 
   sig { returns(String) }
   def inspect
-    "#<#{self.class.name}: #{to_a}>"
+    "#<#{self.class.name}: #{__getobj__}>"
   end
 end
 
@@ -52,11 +52,11 @@ class Requirements < SimpleDelegator
   end
 
   def <<(other)
-    if other.is_a?(Comparable)
-      grep(other.class) do |req|
+    if other.is_a?(Object) && other.is_a?(Comparable)
+      __getobj__.grep(other.class) do |req|
         return self if req > other
 
-        delete(req)
+        __getobj__.delete(req)
       end
     end
     super
@@ -65,6 +65,6 @@ class Requirements < SimpleDelegator
 
   sig { returns(String) }
   def inspect
-    "#<#{self.class.name}: {#{to_a.join(", ")}}>"
+    "#<#{self.class.name}: {#{__getobj__.to_a.join(", ")}}>"
   end
 end
diff --git a/Library/Homebrew/dependencies.rbi b/Library/Homebrew/dependencies.rbi
index 94819cec58..652ce4f99d 100644
--- a/Library/Homebrew/dependencies.rbi
+++ b/Library/Homebrew/dependencies.rbi
@@ -2,6 +2,10 @@
 
 class Dependencies < SimpleDelegator
   include Kernel
+  # This is a workaround to enable `alias eql? ==`
+  # @see https://github.com/sorbet/sorbet/issues/2378#issuecomment-569474238
+  sig { params(arg0: BasicObject).returns(T::Boolean) }
+  def ==(arg0); end
 end
 
 class Requirements < SimpleDelegator
diff --git a/Library/Homebrew/extend/object.rbi b/Library/Homebrew/extend/object.rbi
new file mode 100644
index 0000000000..cc39405568
--- /dev/null
+++ b/Library/Homebrew/extend/object.rbi
@@ -0,0 +1,6 @@
+# typed: strict
+
+class Object
+  sig { returns(T::Boolean) }
+  def present?; end
+end
diff --git a/Library/Homebrew/extend/os/mac/hardware/cpu.rb b/Library/Homebrew/extend/os/mac/hardware/cpu.rb
index 280525c4e7..9e79aa6677 100644
--- a/Library/Homebrew/extend/os/mac/hardware/cpu.rb
+++ b/Library/Homebrew/extend/os/mac/hardware/cpu.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 require "macho"
@@ -114,7 +114,7 @@ module Hardware
         end
       end
 
-      def intel_family
+      def intel_family(_family = nil, _cpu_model = nil)
         case sysctl_int("hw.cpufamily")
         when 0x73d67300 # Yonah: Core Solo/Duo
           :core
diff --git a/Library/Homebrew/extend/os/mac/keg.rb b/Library/Homebrew/extend/os/mac/keg.rb
index 6a47832a5c..6647028c96 100644
--- a/Library/Homebrew/extend/os/mac/keg.rb
+++ b/Library/Homebrew/extend/os/mac/keg.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 class Keg
@@ -108,7 +108,7 @@ class Keg
   # Needed to make symlink permissions consistent on macOS and Linux for
   # reproducible bottles.
   def consistent_reproducible_symlink_permissions!
-    find do |file|
+    path.find do |file|
       File.lchmod 0777, file if file.symlink?
     end
   end
diff --git a/Library/Homebrew/extend/predicable.rb b/Library/Homebrew/extend/predicable.rb
index 914244d5d3..6aa79ad949 100644
--- a/Library/Homebrew/extend/predicable.rb
+++ b/Library/Homebrew/extend/predicable.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 module Predicable
diff --git a/Library/Homebrew/extend/predicable.rbi b/Library/Homebrew/extend/predicable.rbi
new file mode 100644
index 0000000000..cd5e165e4c
--- /dev/null
+++ b/Library/Homebrew/extend/predicable.rbi
@@ -0,0 +1,5 @@
+# typed: strict
+
+module Predicable
+  requires_ancestor { Class }
+end
diff --git a/Library/Homebrew/extend/time.rb b/Library/Homebrew/extend/time.rb
index 1fee2d4bd3..d62de2c8ea 100644
--- a/Library/Homebrew/extend/time.rb
+++ b/Library/Homebrew/extend/time.rb
@@ -1,16 +1,17 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 module TimeRemaining
   refine Time do
     def remaining
+      T.bind(self, Time)
       [0, self - Time.now].max
     end
 
     def remaining!
       r = remaining
 
-      raise Timeout::Error if r <= 0
+      Kernel.raise Timeout::Error if r <= 0
 
       r
     end
diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index 45b671defd..dc5446d160 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 module Homebrew
@@ -161,7 +161,7 @@ module Homebrew
             # specifies the strategy and contains a `strategy` block
             next if (livecheck_strategy != strategy_symbol) || !block_provided
           elsif strategy.const_defined?(:PRIORITY) &&
-                !strategy::PRIORITY.positive? &&
+                !strategy.const_get(:PRIORITY).positive? &&
                 livecheck_strategy != strategy_symbol
             # Ignore strategies with a priority of 0 or lower, unless the
             # strategy is specified in the `livecheck` block
@@ -174,7 +174,7 @@ module Homebrew
         # Sort usable strategies in descending order by priority, using the
         # DEFAULT_PRIORITY when a strategy doesn't contain a PRIORITY constant
         usable_strategies.sort_by do |strategy|
-          (strategy.const_defined?(:PRIORITY) ? -strategy::PRIORITY : -DEFAULT_PRIORITY)
+          (strategy.const_defined?(:PRIORITY) ? -strategy.const_get(:PRIORITY) : -DEFAULT_PRIORITY)
         end
       end
 
@@ -216,7 +216,7 @@ module Homebrew
       # @return [Hash]
       sig { params(url: String, homebrew_curl: T::Boolean).returns(T::Hash[Symbol, T.untyped]) }
       def self.page_content(url, homebrew_curl: false)
-        stderr = nil
+        stderr = T.let(nil, T.nilable(String))
         [:default, :browser].each do |user_agent|
           stdout, stderr, status = curl_with_workarounds(
             *PAGE_CONTENT_CURL_ARGS, url,
diff --git a/Library/Homebrew/utils/pypi.rb b/Library/Homebrew/utils/pypi.rb
index 3fa0277588..c65576a186 100644
--- a/Library/Homebrew/utils/pypi.rb
+++ b/Library/Homebrew/utils/pypi.rb
@@ -1,4 +1,4 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 # Helper functions for updating PyPI resources.
@@ -7,8 +7,6 @@
 module PyPI
   extend T::Sig
 
-  module_function
-
   PYTHONHOSTED_URL_PREFIX = "https://files.pythonhosted.org/packages/"
   private_constant :PYTHONHOSTED_URL_PREFIX
 
@@ -35,13 +33,16 @@ module PyPI
         return
       end
 
-      @name = package_string
-      @name, @version = @name.split("==") if @name.include? "=="
+      if package_string.include? "=="
+        @name, @version = package_string.split("==")
+      else
+        @name = package_string
+      end
 
-      return unless (match = @name.match(/^(.*?)\[(.+)\]$/))
+      return unless (match = T.must(@name).match(/^(.*?)\[(.+)\]$/))
 
       @name = match[1]
-      @extras = match[2].split ","
+      @extras = T.must(match[2]).split ","
     end
 
     # Get name, URL, SHA-256 checksum, and latest version for a given PyPI package.
@@ -87,7 +88,7 @@ module PyPI
 
     sig { params(other: Package).returns(T::Boolean) }
     def same_package?(other)
-      @name.tr("_", "-").casecmp(other.name.tr("_", "-")).zero?
+      T.must(@name.tr("_", "-").casecmp(other.name.tr("_", "-"))).zero?
     end
 
     # Compare only names so we can use .include? and .uniq on a Package array
@@ -109,7 +110,7 @@ module PyPI
   end
 
   sig { params(url: String, version: T.any(String, Version)).returns(T.nilable(String)) }
-  def update_pypi_url(url, version)
+  def self.update_pypi_url(url, version)
     package = Package.new url, is_url: true
 
     return unless package.valid_pypi_package?
@@ -133,8 +134,9 @@ module PyPI
       ignore_non_pypi_packages: T.nilable(T::Boolean),
     ).returns(T.nilable(T::Boolean))
   }
-  def update_python_resources!(formula, version: nil, package_name: nil, extra_packages: nil, exclude_packages: nil,
-                               print_only: false, silent: false, ignore_non_pypi_packages: false)
+  def self.update_python_resources!(formula, version: nil, package_name: nil, extra_packages: nil,
+                                    exclude_packages: nil, print_only: false, silent: false,
+                                    ignore_non_pypi_packages: false)
 
     auto_update_list = formula.tap&.pypi_formula_mappings
     if auto_update_list.present? && auto_update_list.key?(formula.full_name) &&
@@ -282,7 +284,7 @@ module PyPI
     true
   end
 
-  def json_to_packages(json_tree, main_package, exclude_packages)
+  def self.json_to_packages(json_tree, main_package, exclude_packages)
     return [] if json_tree.blank?
 
     json_tree.flat_map do |package_json|