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lambda_calculus.c
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/*
* An untyped lambda calculus [1] interpreter using De Bruijn indices [2] and normal order
* evaluation strategy [3].
*
* [1]: https://en.wikipedia.org/wiki/Lambda_calculus
* [2]: https://en.wikipedia.org/wiki/De_Bruijn_index
* [3]: https://en.wikipedia.org/wiki/Evaluation_strategy#Normal_order
*/
#include <metalang99.h>
// Syntactic terms {
#define var(i) ML99_call(var, i)
#define appl(M, N) ML99_call(appl, M, N)
#define lam(M) ML99_call(lam, M)
#define var_IMPL(i) v(VAR(i))
#define appl_IMPL(M, N) v(APPL(M, N))
#define lam_IMPL(M) v(LAM(M))
#define VAR(i) ML99_CHOICE(var, i)
#define APPL(M, N) ML99_CHOICE(appl, M, N)
#define LAM(M) ML99_CHOICE(lam, M)
// } (Syntactic terms)
// Variable substitution: `M[1=x]` {
#define subst(M, x) ML99_call(subst, M, x)
#define subst_IMPL(M, x) substAux_IMPL(M, x, 1)
#define substAux_IMPL(M, x, depth) ML99_callUneval(ML99_matchWithArgs, M, substAux_, x, depth)
#define substAux_var_IMPL(i, x, depth) \
ML99_IF( \
ML99_NAT_EQ(i, depth), \
v(x), \
ML99_call(ML99_if, ML99_callUneval(ML99_greater, i, depth), v(VAR(ML99_DEC(i)), VAR(i))))
#define substAux_appl_IMPL(M, N, x, depth) \
appl(substAux_IMPL(M, x, depth), substAux_IMPL(N, x, depth))
#define substAux_lam_IMPL(M, x, depth) \
lam(ML99_call(substAux, v(M), incFreeVars_IMPL(x), v(ML99_INC(depth))))
// } (Variable substitution)
// Increment free variables in `M` {
#define incFreeVars(M) ML99_call(incFreeVars, M)
#define incFreeVars_IMPL(M) incFreeVarsAux_IMPL(M, 1)
#define incFreeVarsAux_IMPL(M, depth) ML99_callUneval(ML99_matchWithArgs, M, incFreeVarsAux_, depth)
#define incFreeVarsAux_var_IMPL(i, depth) \
ML99_call(ML99_if, ML99_callUneval(ML99_greaterEq, i, depth), v(VAR(ML99_INC(i)), VAR(i)))
#define incFreeVarsAux_appl_IMPL(M, N, depth) \
appl(incFreeVarsAux_IMPL(M, depth), incFreeVarsAux_IMPL(N, depth))
#define incFreeVarsAux_lam_IMPL(M, depth) lam(incFreeVarsAux_IMPL(M, ML99_INC(depth)))
// } (Increment free variables)
// Evaluation {
#define eval(M) ML99_call(eval, M)
#define eval_IMPL(M) ML99_callUneval(ML99_match, M, eval_)
#define eval_var_IMPL(i) v(VAR(i))
#define eval_appl_IMPL(M, N) ML99_callUneval(ML99_matchWithArgs, M, eval_appl_, N)
#define eval_lam_IMPL(M) lam(eval_IMPL(M))
#define eval_appl_var_IMPL(i, N) appl(v(VAR(i)), eval_IMPL(N))
#define eval_appl_appl_IMPL(M, N, N1) \
ML99_call(ML99_matchWithArgs, eval(appl_IMPL(M, N)), v(eval_appl_appl_, N1))
#define eval_appl_lam_IMPL(M, N) eval(subst_IMPL(M, N))
#define eval_appl_appl_var_IMPL eval_appl_var_IMPL
#define eval_appl_appl_appl_IMPL(M, N, N1) appl(appl_IMPL(M, N), eval_IMPL(N1))
#define eval_appl_appl_lam_IMPL eval_appl_lam_IMPL
// } (Evaluation)
// Syntactical equality {
#define termEq(lhs, rhs) ML99_matchWithArgs(lhs, v(termEq_), rhs)
#define termEq_var_IMPL(i, rhs) termEqPropagate(var, rhs, i)
#define termEq_appl_IMPL(M, N, rhs) termEqPropagate(appl, rhs, M, N)
#define termEq_lam_IMPL(M, rhs) termEqPropagate(lam, rhs, M)
#define termEqPropagate(term_kind, rhs, ...) \
ML99_IF( \
ML99_IDENT_EQ(TERM_, ML99_CHOICE_TAG(rhs), term_kind), \
ML99_matchWithArgs(v(rhs), v(termEq_##term_kind##_), v(__VA_ARGS__)), \
ML99_false())
#define termEq_var_var_IMPL(j, i) v(ML99_NAT_EQ(i, j))
#define termEq_appl_appl_IMPL(M, N, M1, N1) ML99_and(termEq(v(M), v(M1)), termEq(v(N), v(N1)))
#define termEq_lam_lam_IMPL(M, M1) termEq(v(M), v(M1))
#define TERM_var_var ()
#define TERM_appl_appl ()
#define TERM_lam_lam ()
// } (Syntactical equality)
#define ASSERT_REDUCES_TO(lhs, rhs) \
/* Use two interpreter passes: one for `eval(lhs)`, one for `termEq`. Thereby we achieve more \
* Metalang99 reduction steps available. */ \
ML99_ASSERT_UNEVAL(ML99_EVAL(termEq(v(ML99_EVAL(eval(v(lhs)))), v(ML99_EVAL(eval(v(rhs)))))))
// The identity combinator {
#define I LAM(VAR(1))
ASSERT_REDUCES_TO(APPL(I, VAR(5)), VAR(5));
// } (The identity combinator)
// The K, S combinators {
#define K LAM(LAM(VAR(2)))
#define S LAM(LAM(LAM(APPL(APPL(VAR(3), VAR(1)), APPL(VAR(2), VAR(1))))))
ASSERT_REDUCES_TO(APPL(APPL(S, K), K), I);
ASSERT_REDUCES_TO(APPL(APPL(APPL(S, K), S), K), K);
ASSERT_REDUCES_TO(APPL(APPL(APPL(S, K), VAR(5)), VAR(6)), VAR(6));
ASSERT_REDUCES_TO(APPL(APPL(K, VAR(5)), VAR(6)), VAR(5));
// } (The K, S combinators)
// Church booleans {
#define T LAM(LAM(VAR(2)))
#define F LAM(LAM(VAR(1)))
#define NOT LAM(APPL(APPL(VAR(1), F), T))
#define AND LAM(LAM(APPL(APPL(VAR(2), VAR(1)), VAR(2))))
#define OR LAM(LAM(APPL(APPL(VAR(2), VAR(2)), VAR(1))))
#define XOR LAM(LAM(APPL(APPL(VAR(2), APPL(NOT, VAR(1))), VAR(1))))
#define IF LAM(LAM(LAM(APPL(APPL(VAR(3), VAR(2)), VAR(1)))))
ASSERT_REDUCES_TO(APPL(NOT, T), F);
ASSERT_REDUCES_TO(APPL(NOT, F), T);
ASSERT_REDUCES_TO(APPL(NOT, APPL(NOT, T)), T);
ASSERT_REDUCES_TO(APPL(NOT, APPL(NOT, F)), F);
ASSERT_REDUCES_TO(APPL(APPL(AND, T), T), T);
ASSERT_REDUCES_TO(APPL(APPL(AND, T), F), F);
ASSERT_REDUCES_TO(APPL(APPL(AND, F), T), F);
ASSERT_REDUCES_TO(APPL(APPL(AND, F), F), F);
ASSERT_REDUCES_TO(APPL(APPL(OR, T), T), T);
ASSERT_REDUCES_TO(APPL(APPL(OR, T), F), T);
ASSERT_REDUCES_TO(APPL(APPL(OR, F), T), T);
ASSERT_REDUCES_TO(APPL(APPL(OR, F), F), F);
ASSERT_REDUCES_TO(APPL(APPL(XOR, T), T), F);
ASSERT_REDUCES_TO(APPL(APPL(XOR, T), F), T);
ASSERT_REDUCES_TO(APPL(APPL(XOR, F), T), T);
ASSERT_REDUCES_TO(APPL(APPL(XOR, F), F), F);
ASSERT_REDUCES_TO(APPL(APPL(APPL(IF, T), VAR(5)), VAR(6)), VAR(5));
ASSERT_REDUCES_TO(APPL(APPL(APPL(IF, F), VAR(5)), VAR(6)), VAR(6));
// } (Church booleans)
// Church numerals {
#define ZERO LAM(LAM(VAR(1)))
#define SUCC LAM(LAM(LAM(APPL(VAR(2), APPL(APPL(VAR(3), VAR(2)), VAR(1))))))
#define ONE APPL(SUCC, ZERO)
#define TWO APPL(SUCC, ONE)
#define THREE APPL(SUCC, TWO)
#define FOUR APPL(SUCC, THREE)
#define ADD LAM(LAM(LAM(LAM(APPL(APPL(VAR(4), VAR(2)), APPL(APPL(VAR(3), VAR(2)), VAR(1)))))))
#define MUL LAM(LAM(LAM(LAM(APPL(APPL(VAR(4), APPL(VAR(3), VAR(2))), VAR(1))))))
ASSERT_REDUCES_TO(APPL(APPL(ADD, ZERO), ZERO), ZERO);
ASSERT_REDUCES_TO(APPL(APPL(ADD, ZERO), ONE), ONE);
ASSERT_REDUCES_TO(APPL(APPL(ADD, ONE), ZERO), ONE);
ASSERT_REDUCES_TO(APPL(APPL(ADD, ONE), TWO), THREE);
ASSERT_REDUCES_TO(APPL(APPL(MUL, ZERO), ZERO), ZERO);
ASSERT_REDUCES_TO(APPL(APPL(MUL, ZERO), ONE), ZERO);
ASSERT_REDUCES_TO(APPL(APPL(MUL, ONE), ZERO), ZERO);
ASSERT_REDUCES_TO(APPL(APPL(MUL, TWO), TWO), FOUR);
// } (Church numerals)
// Church pairs {
#define PAIR LAM(LAM(LAM(APPL(APPL(VAR(1), VAR(3)), VAR(2)))))
#define FST LAM(APPL(VAR(1), T))
#define SND LAM(APPL(VAR(1), F))
ASSERT_REDUCES_TO(APPL(FST, APPL(APPL(PAIR, VAR(5)), VAR(6))), VAR(5));
ASSERT_REDUCES_TO(APPL(SND, APPL(APPL(PAIR, VAR(5)), VAR(6))), VAR(6));
// } (Church pairs)
// Church lists {
#define NIL F
#define CONS PAIR
#define IS_NIL LAM(APPL(APPL(VAR(1), LAM(LAM(LAM(F)))), T))
#define LIST_1_2_3 APPL(APPL(CONS, VAR(1)), APPL(APPL(CONS, VAR(2)), APPL(APPL(CONS, VAR(3)), NIL)))
ASSERT_REDUCES_TO(APPL(IS_NIL, NIL), T);
ASSERT_REDUCES_TO(APPL(IS_NIL, LIST_1_2_3), F);
// } (Church lists)
// Recursion via self-application (or the Y combinator) is perfectly expressible, though when
// executed, it exhausts the Metalang99 recursion engine limit.
int main(void) {}