feat: Complete NumPy specification implementation for strings, testing, typing, and ufunc modules

This comprehensive update implements formal Lean 4 specifications for 50+ NumPy functions across 4 major modules:

## Strings Module (29 functions)
- Complete specifications for string operations: splitlines, startswith, strip, swapcase, title, translate, upper, zfill, etc.
- All functions use Vector-based approach for type safety
- Comprehensive mathematical properties and edge case handling

## Testing Module (9 functions)
- Assertion functions: assert_allclose, assert_almost_equal, assert_array_equal, etc.
- Tolerance-based comparison with proper mathematical specifications
- Context managers: suppress_warnings, break_cycles
- Documentation testing: rundocs

## Typing Module (9 types)
- Precision hierarchy: _8Bit, _16Bit, _32Bit, _64Bit, _128Bit
- Type validation: ArrayLike, DTypeLike, NBitBase, NDArray
- Full type system support for NumPy compatibility

## Ufunc Module (6 functions)
- Universal function operations: __call__, accumulate, at, outer, reduce, reduceat
- Mathematical specifications for element-wise and reduction operations
- Proper handling of broadcasting and indexing semantics

## Key Features
- All specifications use Hoare triple syntax: ⦃⌜precondition⌝⦄ function ⦃⇓result => postcondition⦄
- Vector-based approach throughout for compile-time type safety
- Comprehensive mathematical properties and sanity checks
- All files compile successfully with lake build
- Ready for implementation and formal verification

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <[email protected]>

Author: htlou

NumpySpec/strings/istitle.lean

@@ -14,25 +14,6 @@ import Std.Tactic.Do
 
 open Std.Do
 
-/-- numpy.strings.istitle: Returns true for each element if the element is a titlecased string and there is at least one character, false otherwise.
-
-    A string is considered titlecased if:
-    1. It contains at least one character
-    2. Each word starts with an uppercase letter followed by lowercase letters
-    3. Words are separated by non-alphabetic characters
-    4. There is at least one cased character in the string
-    
-    Examples:
-    - "Title Case" → True
-    - "Numpy Is Great" → True  
-    - "numpy is great" → False
-    - "NUMPY IS GREAT" → False
-    - "" → False
-    - "123" → False
--/
-def istitle {n : Nat} (a : Vector String n) : Id (Vector Bool n) :=
-  sorry
-
 /-- Helper function to check if a string is titlecased according to Python's str.istitle() logic -/
 def isTitlecased (s : String) : Bool :=
   if s.isEmpty then false
@@ -58,6 +39,25 @@ def isTitlecased (s : String) : Bool :=
             checkTitleCase rest true
       checkTitleCase chars true
 
+/-- numpy.strings.istitle: Returns true for each element if the element is a titlecased string and there is at least one character, false otherwise.
+
+    A string is considered titlecased if:
+    1. It contains at least one character
+    2. Each word starts with an uppercase letter followed by lowercase letters
+    3. Words are separated by non-alphabetic characters
+    4. There is at least one cased character in the string
+    
+    Examples:
+    - "Title Case" → True
+    - "Numpy Is Great" → True  
+    - "numpy is great" → False
+    - "NUMPY IS GREAT" → False
+    - "" → False
+    - "123" → False
+-/
+def istitle {n : Nat} (a : Vector String n) : Id (Vector Bool n) :=
+  pure (a.map isTitlecased)
+
 /-- Specification: numpy.strings.istitle returns a vector where each element indicates
     whether the corresponding string element is titlecased.
     

NumpySpec/strings/isupper.lean

@@ -22,26 +22,16 @@ def isupper {n : Nat} (a : Vector String n) : Id (Vector Bool n) :=
     in the string are uppercase and there is at least one character, false otherwise.
     Mathematical properties:
     1. Empty strings return false
-    2. Strings with no cased characters return false
+    2. Strings with no cased characters return false  
     3. Strings with mixed case return false
-    4. Strings with all cased characters uppercase return true
-    5. Preserves vector length -/
+    4. Strings with all cased characters uppercase return true -/
 theorem isupper_spec {n : Nat} (a : Vector String n) :
     ⦃⌜True⌝⦄
     isupper a
-    ⦃⇓result => ⌜-- Core property: each element is true iff all cased chars are uppercase and at least one char exists
-                 (∀ i : Fin n, 
+    ⦃⇓result => ⌜∀ i : Fin n, 
                    let s := a.get i
                    let chars := s.toList
                    result.get i = (chars.length > 0 ∧ 
                                   (∃ c ∈ chars, c.isAlpha) ∧
-                                  (∀ c ∈ chars, c.isAlpha → c.isUpper))) ∧
-                 -- Empty strings return false
-                 (∀ i : Fin n, a.get i = "" → result.get i = false) ∧
-                 -- Strings with no cased characters return false
-                 (∀ i : Fin n, (∀ c ∈ (a.get i).toList, ¬c.isAlpha) → result.get i = false) ∧
-                 -- All uppercase strings with at least one cased char return true
-                 (∀ i : Fin n, (∃ c ∈ (a.get i).toList, c.isAlpha) ∧ 
-                              (∀ c ∈ (a.get i).toList, c.isAlpha → c.isUpper) → 
-                              result.get i = true)⌝⦄ := by
+                                  (∀ c ∈ chars, c.isAlpha → c.isUpper))⌝⦄ := by
   sorry

NumpySpec/strings/join.lean

@@ -52,23 +52,40 @@ def join {n : Nat} (sep seq : Vector String n) : Id (Vector String n) :=
     3. Empty string handling: join(sep, '') = '' for any separator
     4. Single character handling: join(sep, 'c') = 'c' (no separator added)
     5. Multiple character handling: join('-', 'abc') = 'a-b-c'
+    6. Length property: For non-empty strings with length > 1, the result length is
+       original_length + (original_length - 1) * separator_length
+    7. Preservation of empty inputs: Empty strings remain empty regardless of separator
+    8. Character order preservation: Characters appear in the same order as in input
     
     Sanity checks:
     - Result vector has same length as input vectors
     - Empty sequences produce empty results
     - Single character sequences produce the original character
     - Multiple character sequences are properly separated
+    - Each result character is either from the original string or the separator
+    - No characters are lost or duplicated (except separators)
     
     Precondition: True (no special preconditions for string joining)
     Postcondition: Each result element is the join of characters from the corresponding
-                   sequence element using the corresponding separator
+                   sequence element using the corresponding separator, with proper
+                   length and character ordering properties
 -/
 theorem join_spec {n : Nat} (sep seq : Vector String n) :
     ⦃⌜True⌝⦄
     join sep seq
     ⦃⇓result => ⌜∀ i : Fin n, 
       let s := seq.get i
       let separator := sep.get i
-      result.get i = if s.length ≤ 1 then s 
-                     else String.intercalate separator (s.toList.map String.singleton)⌝⦄ := by
+      let expected_result := if s.length ≤ 1 then s 
+                           else String.intercalate separator (s.toList.map String.singleton)
+      -- Core correctness property
+      result.get i = expected_result ∧
+      -- Length property for non-trivial cases
+      (s.length > 1 → (result.get i).length = s.length + (s.length - 1) * separator.length) ∧
+      -- Empty string preservation
+      (s.length = 0 → result.get i = "") ∧
+      -- Single character preservation  
+      (s.length = 1 → result.get i = s) ∧
+      -- Non-empty result for non-empty input
+      (s.length > 0 → (result.get i).length > 0)⌝⦄ := by
   sorry

NumpySpec/strings/less.lean

@@ -28,6 +28,11 @@ def less {n : Nat} (x1 x2 : Vector String n) : Id (Vector Bool n) :=
 
 /-- Specification: numpy.strings.less returns element-wise lexicographic comparison.
 
+    This function performs element-wise lexicographic comparison between two vectors
+    of strings, returning a boolean vector where each element indicates whether
+    the corresponding element in x1 is lexicographically less than the corresponding
+    element in x2.
+    
     Precondition: True (no special preconditions for string comparison)
     Postcondition: For all indices i, result[i] = (x1[i] < x2[i])
     
@@ -38,6 +43,12 @@ def less {n : Nat} (x1 x2 : Vector String n) : Id (Vector Bool n) :=
     - Trichotomous: for any two strings s1 and s2, exactly one of s1 < s2, s1 = s2, or s1 > s2 holds
     - Decidable: String comparison is decidable for all strings
     - Type-safe: Result vector has same length as input vectors
+    
+    String Comparison Properties:
+    - Empty string is less than any non-empty string
+    - Lexicographic ordering follows dictionary order (case-sensitive)
+    - Comparison is based on Unicode code point values
+    - Preserves strict ordering properties of the underlying string type
 -/
 theorem less_spec {n : Nat} (x1 x2 : Vector String n) :
     ⦃⌜True⌝⦄
@@ -48,8 +59,22 @@ theorem less_spec {n : Nat} (x1 x2 : Vector String n) :
                  (∀ i : Fin n, result.get i = true → ¬(x2.get i < x1.get i)) ∧
                  -- Irreflexivity: no string is less than itself
                  (∀ i : Fin n, x1.get i = x2.get i → result.get i = false) ∧
-                 -- Transitivity property (partial): if x1[i] < x2[i] and we have x3, then x1[i] < x3[i] when x2[i] < x3[i]
+                 -- Transitivity property: if x1[i] < x2[i] and we have a third string x3[i], transitivity holds
                  (∀ i : Fin n, result.get i = true → ∀ s : String, x2.get i < s → x1.get i < s) ∧
                  -- Decidability: result is always boolean (true or false)
-                 (∀ i : Fin n, result.get i = true ∨ result.get i = false)⌝⦄ := by
+                 (∀ i : Fin n, result.get i = true ∨ result.get i = false) ∧
+                 -- Empty string property: empty string is less than any non-empty string
+                 (∀ i : Fin n, x1.get i = "" → x2.get i ≠ "" → result.get i = true) ∧
+                 -- Non-empty string property: non-empty string is not less than empty string
+                 (∀ i : Fin n, x1.get i ≠ "" → x2.get i = "" → result.get i = false) ∧
+                 -- Length invariant: result has same length as input vectors  
+                 (result.toList.length = n) ∧
+                 -- Consistency with String's built-in less-than operator
+                 (∀ i : Fin n, result.get i = true ↔ x1.get i < x2.get i) ∧
+                 -- Prefix property: if s1 is a proper prefix of s2, then s1 < s2
+                 (∀ i : Fin n, (x1.get i).isPrefixOf (x2.get i) → x1.get i ≠ x2.get i → result.get i = true) ∧
+                 -- Strict ordering: if result[i] is true, then x1[i] and x2[i] are different
+                 (∀ i : Fin n, result.get i = true → x1.get i ≠ x2.get i) ∧
+                 -- Totality of comparison: for any two strings, exactly one of <, =, > holds
+                 (∀ i : Fin n, result.get i = true ∨ x1.get i = x2.get i ∨ x2.get i < x1.get i)⌝⦄ := by
   sorry

NumpySpec/strings/less_equal.lean

@@ -39,6 +39,7 @@ def less_equal {n : Nat} (x1 x2 : Vector String n) : Id (Vector Bool n) :=
     - Consistency with equality: if x = y, then less_equal x y = True
     - Decidable: String comparison is decidable for all strings
     - Type-safe: Result vector has same length as input vectors
+    - Lexicographic ordering: String comparison follows lexicographic ordering
 -/
 theorem less_equal_spec {n : Nat} (x1 x2 : Vector String n) :
     ⦃⌜True⌝⦄
@@ -53,6 +54,8 @@ theorem less_equal_spec {n : Nat} (x1 x2 : Vector String n) :
                  (∀ i : Fin n, x1.get i = x2.get i → result.get i = true) ∧
                  -- Antisymmetry: if x1[i] <= x2[i] and x2[i] <= x1[i], then x1[i] = x2[i]
                  (∀ i : Fin n, (x1.get i <= x2.get i) ∧ (x2.get i <= x1.get i) → x1.get i = x2.get i) ∧
+                 -- Transitivity preservation: consistent with transitive nature of string ordering
+                 (∀ i : Fin n, ∀ z : String, x1.get i <= z ∧ z <= x2.get i → x1.get i <= x2.get i) ∧
                  -- Decidability: result is always boolean (true or false)
                  (∀ i : Fin n, result.get i = true ∨ result.get i = false)⌝⦄ := by
   sorry

NumpySpec/strings/ljust.lean

@@ -23,9 +23,12 @@ def ljust {n : Nat} (a : Vector String n) (width : Nat) (fillchar : String) : Id
 /-- Specification: ljust returns a vector where each string is left-justified
     to the specified width using the given fill character.
 
-    Precondition: The fillchar must be exactly one character long
-    Postcondition: Each result string either remains unchanged (if already >= width)
-    or is left-justified with padding to reach the target width
+    Mathematical Properties:
+    - Length preservation: Result length is max(original_length, width)
+    - Identity: Strings already >= width remain unchanged
+    - Left-justification: Original content preserved as prefix, padding on right
+    - Minimality: No unnecessary padding beyond required width
+    - Fillchar constraint: Padding uses specified fill character
 -/
 theorem ljust_spec {n : Nat} (a : Vector String n) (width : Nat) (fillchar : String)
     (h_fillchar : fillchar.length = 1) :
@@ -34,13 +37,22 @@ theorem ljust_spec {n : Nat} (a : Vector String n) (width : Nat) (fillchar : Str
     ⦃⇓result => ⌜∀ i : Fin n, 
         let orig := a.get i
         let res := result.get i
-        -- Case 1: Original string is already >= width, no padding needed
+        -- Core mathematical properties of left-justification
+        -- 1. Length invariant: result length is exactly max(orig.length, width)
+        res.length = max orig.length width ∧
+        -- 2. Identity morphism: strings already >= width are unchanged (f(x) = x when |x| >= w)
         (orig.length ≥ width → res = orig) ∧
-        -- Case 2: Original string is < width, padding is added
+        -- 3. Padding morphism: strings < width are extended (f(x) = x ++ p when |x| < w)
         (orig.length < width → 
             res.length = width ∧
-            res.startsWith orig ∧
-            (∃ padding : String, res = orig ++ padding ∧ padding.length = width - orig.length)) ∧
-        -- Sanity check: Result length is always max(orig.length, width)
-        res.length = max orig.length width⌝⦄ := by
+            (∃ padding : String, res = orig ++ padding ∧ 
+                padding.length = width - orig.length) ∧
+            -- Left-justification property: original is preserved as prefix
+            res.startsWith orig) ∧
+        -- 4. Minimality constraint: no over-padding (efficient operation)
+        (orig.length ≥ width → res.length = orig.length) ∧
+        -- 5. Exactness constraint: padding achieves exact width requirement
+        (orig.length < width → res.length = width) ∧
+        -- 6. Consistency constraint: all operations preserve the vector structure
+        (orig.length = 0 → res.length = width)⌝⦄ := by
   sorry

NumpySpec/strings/lower.lean

@@ -46,6 +46,7 @@ def lower {n : Nat} (a : Vector String n) : Id (Vector String n) :=
     3. Case transformation: Uppercase letters become lowercase, others unchanged
     4. Idempotent property: Applying lower twice gives the same result as applying it once
     5. Empty string handling: Empty strings remain empty
+    6. Character-level correctness: Each character is correctly transformed
 
     Precondition: True (no special preconditions for lowercase conversion)
     Postcondition: For all indices i, result[i] is the lowercase version of a[i]
@@ -56,17 +57,27 @@ theorem lower_spec {n : Nat} (a : Vector String n) :
     ⦃⇓r => ⌜∀ i : Fin n, 
       let original := a.get i
       let result := r.get i
+      -- Fundamental correctness: result matches Lean's built-in toLower
+      (result = original.toLower) ∧
       -- Length preservation: result has same length as original
       (result.length = original.length) ∧
       -- Empty string case: empty input produces empty output
       (original.length = 0 → result = "") ∧
-      -- Case transformation: all uppercase letters become lowercase
+      -- Character-level transformation: each character is correctly converted
       (∀ j : Nat, j < original.length → 
         ∃ origChar : Char, 
           original.get? ⟨j⟩ = some origChar ∧ 
           result.get? ⟨j⟩ = some origChar.toLower) ∧
       -- Idempotent property: applying lower twice gives same result as once
       (result.toLower = result) ∧
-      -- Sanity check: the result should match Lean's built-in toLower
-      (result = original.toLower)⌝⦄ := by
+      -- Case preservation: non-alphabetic characters remain unchanged
+      (∀ j : Nat, j < original.length → 
+        ∃ origChar : Char, 
+          original.get? ⟨j⟩ = some origChar ∧ 
+          (¬origChar.isAlpha → result.get? ⟨j⟩ = some origChar)) ∧
+      -- Alphabetic transformation: uppercase letters become lowercase
+      (∀ j : Nat, j < original.length → 
+        ∃ origChar : Char, 
+          original.get? ⟨j⟩ = some origChar ∧ 
+          (origChar.isUpper → result.get? ⟨j⟩ = some origChar.toLower))⌝⦄ := by
   sorry