Extract built-in tool spec constructors into codex-tools (#16493)

## Why

`core/src/tools/spec.rs` still had a few built-in tool specs assembled
inline even though those definitions are pure metadata and already live
conceptually in `codex-tools`. Keeping that construction in `codex-core`
makes `spec.rs` do more than registry orchestration and slows the
migration toward a right-sized `codex-tools` crate.

This continues the extraction stack from #16379, #16471, #16477, #16481,
and #16482.

## What Changed

- added `create_local_shell_tool()`, `create_web_search_tool(...)`, and
`create_image_generation_tool(...)` to `codex-rs/tools/src/tool_spec.rs`
- exported those helpers from `codex-rs/tools/src/lib.rs`
- switched `codex-rs/core/src/tools/spec.rs` to call those helpers
instead of constructing `ToolSpec::LocalShell`, `ToolSpec::WebSearch`,
and `ToolSpec::ImageGeneration` inline
- removed the remaining core-local web-search content-type constant and
made the affected spec test assert the literal expected values directly

This is intended to be a straight refactor: tool behavior and wire shape
should not change.

## Testing

- `cargo test -p codex-tools`
- `cargo test -p codex-core tools::spec::tests`

codex-rs/tools/src/lib.rs

@@ -102,7 +102,11 @@ pub use tool_spec::ConfiguredToolSpec;
 pub use tool_spec::ResponsesApiWebSearchFilters;
 pub use tool_spec::ResponsesApiWebSearchUserLocation;
 pub use tool_spec::ToolSpec;
+pub use tool_spec::WebSearchToolOptions;
+pub use tool_spec::create_image_generation_tool;
+pub use tool_spec::create_local_shell_tool;
 pub use tool_spec::create_tools_json_for_responses_api;
+pub use tool_spec::create_web_search_tool;
 pub use utility_tool::create_list_dir_tool;
 pub use utility_tool::create_test_sync_tool;
 pub use view_image::ViewImageToolOptions;

codex-rs/tools/src/tool_spec.rs

@@ -1,13 +1,18 @@
 use crate::FreeformTool;
 use crate::JsonSchema;
 use crate::ResponsesApiTool;
+use codex_protocol::config_types::WebSearchConfig;
 use codex_protocol::config_types::WebSearchContextSize;
 use codex_protocol::config_types::WebSearchFilters as ConfigWebSearchFilters;
+use codex_protocol::config_types::WebSearchMode;
 use codex_protocol::config_types::WebSearchUserLocation as ConfigWebSearchUserLocation;
 use codex_protocol::config_types::WebSearchUserLocationType;
+use codex_protocol::openai_models::WebSearchToolType;
 use serde::Serialize;
 use serde_json::Value;
 
+const WEB_SEARCH_TEXT_AND_IMAGE_CONTENT_TYPES: [&str; 2] = ["text", "image"];
+
 /// When serialized as JSON, this produces a valid "Tool" in the OpenAI
 /// Responses API.
 #[derive(Debug, Clone, Serialize, PartialEq)]
@@ -61,6 +66,54 @@ impl ToolSpec {
     }
 }
 
+pub fn create_local_shell_tool() -> ToolSpec {
+    ToolSpec::LocalShell {}
+}
+
+pub fn create_image_generation_tool(output_format: &str) -> ToolSpec {
+    ToolSpec::ImageGeneration {
+        output_format: output_format.to_string(),
+    }
+}
+
+pub struct WebSearchToolOptions<'a> {
+    pub web_search_mode: Option<WebSearchMode>,
+    pub web_search_config: Option<&'a WebSearchConfig>,
+    pub web_search_tool_type: WebSearchToolType,
+}
+
+pub fn create_web_search_tool(options: WebSearchToolOptions<'_>) -> Option<ToolSpec> {
+    let external_web_access = match options.web_search_mode {
+        Some(WebSearchMode::Cached) => Some(false),
+        Some(WebSearchMode::Live) => Some(true),
+        Some(WebSearchMode::Disabled) | None => None,
+    }?;
+
+    let search_content_types = match options.web_search_tool_type {
+        WebSearchToolType::Text => None,
+        WebSearchToolType::TextAndImage => Some(
+            WEB_SEARCH_TEXT_AND_IMAGE_CONTENT_TYPES
+                .into_iter()
+                .map(str::to_string)
+                .collect(),
+        ),
+    };
+
+    Some(ToolSpec::WebSearch {
+        external_web_access: Some(external_web_access),
+        filters: options
+            .web_search_config
+            .and_then(|config| config.filters.clone().map(Into::into)),
+        user_location: options
+            .web_search_config
+            .and_then(|config| config.user_location.clone().map(Into::into)),
+        search_context_size: options
+            .web_search_config
+            .and_then(|config| config.search_context_size),
+        search_content_types,
+    })
+}
+
 #[derive(Debug, Clone, PartialEq)]
 pub struct ConfiguredToolSpec {
     pub spec: ToolSpec,