1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
|
/*
* 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) {}
|
