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+/* Turing Completeness: Lambda Calculus Foundations */
+/* Demonstrates: Church encodings, combinators, functional completeness */
+
+..out "=== Lambda Calculus Foundations Test ===";
+
+/* Test 1: Church Numerals (encoding numbers as functions) */
+/* Church 0: f -> x -> x */
+church_zero : f x -> x;
+
+/* Church 1: f -> x -> f x */
+church_one : f x -> f x;
+
+/* Church 2: f -> x -> f (f x) */  
+church_two : f x -> f (f x);
+
+/* Successor function: n -> f -> x -> f (n f x) */
+church_succ : n f x -> f (n f x);
+
+/* Test successor */
+church_three : church_succ church_two;
+
+/* Convert church numeral to integer for testing */
+inc : x -> x + 1;
+three_as_int : church_three @inc 0;
+
+..assert three_as_int = 3;
+..out "1. Church numerals: PASS";
+
+/* Test 2: Church Booleans */
+church_true : x y -> x;
+church_false : x y -> y;
+
+/* Church conditional: p -> x -> y -> p x y */
+church_if : p x y -> p x y;
+
+/* Test boolean logic */
+bool_test1 : church_if @church_true "yes" "no";
+bool_test2 : church_if @church_false "yes" "no"; 
+
+..assert bool_test1 = "yes";
+..assert bool_test2 = "no";
+..out "2. Church booleans: PASS";
+
+/* Test 3: Combinators (S, K, I) */
+/* S combinator: f -> g -> x -> f x (g x) */
+s_combinator : f g x -> f x (g x);
+
+/* K combinator: x -> y -> x */  
+k_combinator : x y -> x;
+
+/* I combinator: x -> x (can be derived as S K K) */
+i_combinator : @s_combinator @k_combinator @k_combinator;
+
+/* Test I combinator */
+identity_test : i_combinator 42;
+
+..assert identity_test = 42;
+..out "3. SKI combinators: PASS";
+
+/* Test 4: Y Combinator (fixed point for recursion) */
+/* Simplified recursive implementation without nested lambdas */
+y_helper : f x -> f (x x);
+y_inner : f x -> y_helper f x;
+y_comb : f -> y_helper f @y_inner;
+
+/* Factorial using Y combinator */
+fact_func : f n -> when n is 0 then 1 _ then n * (f (n - 1));
+y_factorial : @y_comb @fact_func;
+
+factorial_result : y_factorial 5;
+
+..assert factorial_result = 120;
+..out "4. Y combinator recursion: PASS";
+
+/* Test 5: Currying and partial application */
+curry : f x y -> f x y;
+add_func : x y -> x + y;
+add_curry : curry @add_func;
+add_five : add_curry 5;
+
+curried_result : add_five 10;
+
+..assert curried_result = 15;
+..out "5. Currying: PASS";
+
+..out "✅ All lambda calculus tests passed";
+..out "Demonstrates: Functional completeness, Church encodings, combinatorial logic";
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