Add then and orelse rewrite combinators and tests

src/library/rewrite/fo_match.cpp

@@ -5,6 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Soonho Kong
 */
 #include <utility>
+#include "util/trace.h"
 #include "kernel/expr.h"
 #include "kernel/context.h"
 #include "library/all/all.h"
@@ -30,64 +31,45 @@ std::ostream & operator<<(std::ostream & out, subst_map & s) {
 }
 
 bool fo_match::match_var(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_var : ("
+    lean_trace("fo_match", tout << "match_var : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
-
     unsigned idx = var_idx(p);
     if (idx < o) {
         // Current variable is the one created by lambda inside of pattern
-        // and it is not a target of pattern matching.
+        // and it is *not* a target of pattern matching.
         return p == t;
     } else {
-        auto it = s.find(p);
+        auto it = s.find(idx);
         if (it != s.end()) {
             // This variable already has an entry in the substitution
             // map. We need to make sure that 't' and s[idx] are the
             // same
-            cout << "match_var exist:" << p << " |-> " << it->second << endl;
+            lean_trace("fo_match", tout << "match_var exist:" << idx << " |-> " << it->second << endl;);
             return it->second == t;
         }
         // This variable has no entry in the substituition map. Let's
         // add one.
-        s.insert(std::make_pair(p, t));
+        s.insert(std::make_pair(idx, t));
-        cout << "match_var MATCHED : " << s << endl;
+        lean_trace("fo_match", tout << "match_var MATCHED : " << s << endl;);
         return true;
     }
 }
 
 bool fo_match::match_constant(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_constant : ("
+    lean_trace("fo_match", tout << "match_constant : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
     return p == t;
 }
 
 bool fo_match::match_value(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_value : ("
+    lean_trace("fo_match", tout << "match_value : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
     return p == t;
 }
 
 bool fo_match::match_app(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_app : ("
+    lean_trace("fo_match", tout << "match_app : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
+    if (!is_app(t))
-         << t << ", "
+        return false;
-         << o << ", "
-         << s << ")"
-         << endl;
     unsigned num_p = num_args(p);
-    unsigned num_t = num_args(p);
+    unsigned num_t = num_args(t);
     if (num_p != num_t) {
         return false;
     }
@@ -100,15 +82,8 @@ bool fo_match::match_app(expr const & p, expr const & t, unsigned o, subst_map &
 }
 
 bool fo_match::match_lambda(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_lambda : ("
+    lean_trace("fo_match", tout << "match_lambda : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
+    lean_trace("fo_match", tout << "fun (" << abst_name(p) << " : " << abst_domain(p) << "), " << abst_body(p) << endl;);
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
-    cout << "fun (" << abst_name(p)
-         << " : " << abst_domain(p)
-         << "), " << abst_body(p) << endl;
     if (!is_lambda(t)) {
         return false;
     } else {
@@ -126,16 +101,8 @@ bool fo_match::match_lambda(expr const & p, expr const & t, unsigned o, subst_ma
 }
 
 bool fo_match::match_pi(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_pi : ("
+    lean_trace("fo_match", tout << "match_pi : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
+    lean_trace("fo_match", tout << "Pi (" << abst_name(p) << " : " << abst_domain(p) << "), " << abst_body(p) << endl;);
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
-    cout << "Pi (" << abst_name(p)
-         << " : " << abst_domain(p)
-         << "), " << abst_body(p) << endl;
-
     if (!is_pi(t)) {
         return false;
     } else {
@@ -153,33 +120,19 @@ bool fo_match::match_pi(expr const & p, expr const & t, unsigned o, subst_map &
 }
 
 bool fo_match::match_type(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_type : ("
+    lean_trace("fo_match", tout << "match_type : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
     return p == t;
 }
 
 bool fo_match::match_eq(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_eq : ("
+    lean_trace("fo_match", tout << "match_eq : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
+    if (!is_eq(t))
-         << t << ", "
+        return false;
-         << o << ", "
-         << s << ")"
-         << endl;
     return match(eq_lhs(p), eq_lhs(t), o, s) && match(eq_rhs(p), eq_rhs(t), o, s);
 }
 
 bool fo_match::match_let(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_let : ("
+    lean_trace("fo_match", tout << "match_let : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
-
     if (!is_let(t)) {
         return false;
     } else {
@@ -202,23 +155,12 @@ bool fo_match::match_let(expr const & p, expr const & t, unsigned o, subst_map &
     }
 }
 bool fo_match::match_metavar(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match_meta : ("
+    lean_trace("fo_match", tout << "match_meta : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
     return p == t;
 }
 
 bool fo_match::match(expr const & p, expr const & t, unsigned o, subst_map & s) {
-    cout << "match : ("
+    lean_trace("fo_match", tout << "match : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
-         << p << ", "
-         << t << ", "
-         << o << ", "
-         << s << ")"
-         << endl;
-
     switch (p.kind()) {
     case expr_kind::Var:
         return match_var(p, t, o, s);

src/library/rewrite/fo_match.h

@@ -17,7 +17,7 @@ Author: Soonho Kong
 
 namespace lean {
 
-typedef expr_map<expr> subst_map;
+using subst_map = std::unordered_map<unsigned, expr>;
 
 class fo_match {
 private:

src/library/rewrite/rewrite.cpp

@@ -1,13 +1,16 @@
 /*
-Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+  Copyright (c) 2013 Microsoft Corporation. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
+  Released under Apache 2.0 license as described in the file LICENSE.
 
-Author: Soonho Kong
+  Author: Soonho Kong
 */
+#include "util/trace.h"
 #include "kernel/abstract.h"
 #include "kernel/builtin.h"
 #include "kernel/context.h"
 #include "kernel/environment.h"
+#include "kernel/replace.h"
+#include "library/basic_thms.h"
 #include "library/printer.h"
 #include "library/rewrite/fo_match.h"
 #include "library/rewrite/rewrite.h"
@@ -27,51 +30,85 @@ Author: Soonho Kong
 
 using std::cout;
 using std::endl;
+using std::pair;
+using std::make_pair;
 
 namespace lean {
 
-theorem_rw::theorem_rw(expr const & t, expr const & v)
+theorem_rewrite::theorem_rewrite(expr const & type, expr const & body)
-    : thm_t(t), thm_v(v), num_args(0) {
+    : thm_type(type), thm_body(body), num_args(0) {
-    cout << "================= Theorem Rewrite Constructor Start ===================" << endl;
+    lean_trace("rewrite", tout << "Type = " << thm_type << endl;);
-    cout << "Type = " << thm_t << endl;
+    lean_trace("rewrite", tout << "Body = " << thm_body << endl;);
-    cout << "Body = " << thm_v << endl;
 
     // We expect the theorem is in the form of
-    // Pi (x_1 : t_1 ... x_n : t_n), t = s
+    // Pi (x_1 : t_1 ... x_n : t_n), pattern = rhs
-    expr tmp = t;
+    pattern = type;
-    while (is_pi(tmp)) {
+    while (is_pi(pattern)) {
-        tmp = abst_body(tmp);
+        pattern = abst_body(pattern);
         num_args++;
     }
-    if (!is_eq(tmp)) {
+    if (!is_eq(pattern)) {
-        cout << "Theorem " << t << " is not in the form of "
+        lean_trace("rewrite", tout << "Theorem " << thm_type << " is not in the form of "
-                  << "Pi (x_1 : t_1 ... x_n : t_n), t = s" << endl;
+                   << "Pi (x_1 : t_1 ... x_n : t_n), pattern = rhs" << endl;);
     }
-    cout << "OK. Number of Arg = " << num_args << endl;
+    rhs = eq_rhs(pattern);
-    cout << "================= Theorem Rewrite Constructor END ===================" << endl;
+    pattern = eq_lhs(pattern);
+
+    lean_trace("rewrite", tout << "Number of Arg = " << num_args << endl;);
 }
 
-void theorem_rw::operator()(context & ctx, expr const & t) {
+pair<expr, expr> theorem_rewrite::operator()(context & ctx, expr const & v, expr const & ) const throw(rewrite_exception) {
-    cout << "================= Theorem Rewrite () START ===================" << endl;
+    lean_trace("rewrite", tout << "Context = " << ctx << endl;);
-    cout << "Context = " << ctx << endl;
+    lean_trace("rewrite", tout << "Term = " << v << endl;);
-    cout << "Term = " << t << endl;
+    lean_trace("rewrite", tout << "Pattern = " << pattern << endl;);
-    expr tmp = thm_t;
+    lean_trace("rewrite", tout << "Num Args = " << num_args << endl;);
-    while (is_pi(tmp)) {
+
-        tmp = abst_body(tmp);
-        num_args++;
-    }
-    if (!is_eq(tmp)) {
-        cout << "Theorem " << t << " is not in the form of "
-                  << "Pi (x_1 : t_1 ... x_n : t_n), t = s" << endl;
-    }
-    expr const & lhs = eq_lhs(tmp);
-    expr const & rhs = eq_rhs(tmp);
     fo_match fm;
     subst_map s;
-    fm.match(lhs, t, 0, s);
+    if (!fm.match(pattern, v, 0, s)) {
+        throw rewrite_exception();
+    }
+
     // apply s to rhs
+    auto f = [&s](expr const & e, unsigned offset) -> expr {
+        if (is_var(e)) {
+            lean_trace("rewrite", tout << "Inside of apply : offset = " << offset
+                       << ", e = " << e
+                       << ", idx = " << var_idx(e)  << endl;);
+            unsigned idx = var_idx(e);
+            auto it = s.find(idx);
+            if (it != s.end()) {
+                lean_trace("rewrite", tout << "Inside of apply : s[" << idx << "] = " << s[idx] << endl;);
+                return s[idx];
+            }
+        }
+        return e;
+    };
 
-    cout << "================= Theorem Rewrite () END ===================" << endl;
+    expr new_rhs = replace_fn<decltype(f)>(f)(rhs);
+    lean_trace("rewrite", tout << "New RHS = " << new_rhs << endl;);
+
+    expr proof = thm_body;
+    for (int i = num_args -1 ; i >= 0; i--) {
+        proof = mk_app(proof, s[i]);
+        lean_trace("rewrite", tout << "proof: " << i << "\t" << s[i] << "\t" << proof << endl;);
+    }
+    lean_trace("rewrite", tout << "Proof   = " << proof << endl;);
+    return make_pair(new_rhs, proof);
 }
 
+pair<expr, expr> orelse_rewrite::operator()(context & ctx, expr const & v, expr const & t) const throw(rewrite_exception) {
+    try {
+        return rewrite1(ctx, v, t);
+    } catch (rewrite_exception & ) {
+        return rewrite2(ctx, v, t);
+    }
+}
+
+pair<expr, expr> then_rewrite::operator()(context & ctx, expr const & v, expr const & t) const throw(rewrite_exception) {
+    pair<expr, expr> result1 = rewrite1(ctx, v, t);
+    pair<expr, expr> result2 = rewrite2(ctx, result1.first, t);
+    return make_pair(result2.first,
+                     Trans(t, v, result1.first, result2.first, result1.second, result2.second));
+}
 }

src/library/rewrite/rewrite.h

@@ -5,17 +5,56 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Soonho Kong
 */
 #pragma once
+#include <utility>
+#include "util/exception.h"
 
 namespace lean {
 
-class theorem_rw {
+class rewrite_exception : public exception {
+};
+
+class rewrite {
+public:
+    virtual std::pair<expr, expr> operator()(context & ctx, expr const & v, expr const & t) const = 0;
+};
+
+class theorem_rewrite : public rewrite {
 private:
-    expr const & thm_t;
+    expr const & thm_type;
-    expr const & thm_v;
+    expr const & thm_body;
+    expr pattern;
+    expr rhs;
     unsigned num_args;
 
 public:
-    theorem_rw(expr const & t, expr const & v);
+    theorem_rewrite(expr const & type, expr const & body);
-    void operator()(context & ctx, expr const & t);
+    std::pair<expr, expr> operator()(context & ctx, expr const & v, expr const & t) const throw(rewrite_exception);
+};
+
+class orelse_rewrite : public rewrite {
+private:
+    rewrite const & rewrite1;
+    rewrite const & rewrite2;
+public:
+    orelse_rewrite(rewrite const & rw1, rewrite const & rw2) :
+        rewrite1(rw1), rewrite2(rw2) { }
+    std::pair<expr, expr> operator()(context & ctx, expr const & v, expr const & t) const throw(rewrite_exception);
+};
+
+class then_rewrite : public rewrite {
+private:
+    rewrite const & rewrite1;
+    rewrite const & rewrite2;
+public:
+    then_rewrite(rewrite const & rw1, rewrite const & rw2) :
+        rewrite1(rw1), rewrite2(rw2) { }
+    std::pair<expr, expr> operator()(context & ctx, expr const & v, expr const & t) const throw(rewrite_exception);
+};
+
+class fail_rewrite : public rewrite {
+public:
+    std::pair<expr, expr> operator()(context &, expr const &) const throw(rewrite_exception) {
+        throw rewrite_exception();
+    }
 };
 }

src/tests/library/rewrite/rewrite.cpp

@@ -4,42 +4,247 @@ Released under Apache 2.0 license as described in the file LICENSE.
 
 Author: Soonho Kong
 */
+#include "util/trace.h"
 #include "kernel/abstract.h"
 #include "kernel/context.h"
 #include "kernel/expr.h"
+#include "kernel/type_checker.h"
 #include "library/all/all.h"
 #include "library/arith/arith.h"
 #include "library/arith/nat.h"
 #include "library/rewrite/fo_match.h"
 #include "library/rewrite/rewrite.h"
+#include "library/basic_thms.h"
 #include "library/printer.h"
 using namespace lean;
 
 using std::cout;
+using std::pair;
 using std::endl;
 
-int main() {
+static void theorem_rewrite1_tst() {
+    cout << "=== theorem_rewrite1_tst() ===" << endl;
+    // Theorem:     Pi(x y : N), x + y = y + x := ADD_COMM x y
+    // Term   :     a + b
+    // Result :     (b + a, ADD_COMM a b)
+    expr a        = Const("a");                  // a  : Nat
+    expr b        = Const("b");                  // b  : Nat
+    expr a_plus_b = nAdd(a, b);
+    expr b_plus_a = nAdd(b, a);
+    expr add_comm_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+    expr add_comm_thm_body = Const("ADD_COMM");
+
     environment env = mk_toplevel();
-    env.add_var("x", Nat);
+    env.add_var("a", Nat);
-    expr x    = Const("x");                  // x  : Nat
+    env.add_var("b", Nat);
-    expr y    = Const("y");                  // y  : Nat
+    env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
-    expr zero = nVal(0);                     // 0  : Nat
-    expr x_plus_zero = nAdd(x, zero);        // x_plus_zero := x + 0
-    expr y_plus_zero = nAdd(y, zero);        // y_plus_zero  := y + 0
-    cout << "x := " << x << endl;
-    cout << "y := " << y << endl;
-    cout << "x + 0 := " << x_plus_zero << endl;
-    cout << "x + 0 := " << y_plus_zero << endl;
-    //env.display(cout);
 
-    expr thm_t = Pi("x", Nat, Eq(nAdd(Const("x"), nVal(0)), Const("x"))); // Pi (x : Z), x + 0 = x
+    // Rewriting
-    cout << "theorem := " << thm_t << endl;
+    theorem_rewrite add_comm_thm_rewriter(add_comm_thm_type, add_comm_thm_body);
-    env.add_axiom("H1", thm_t); // H1 : Pi (x : N), x = x + 0
-    expr thm_v = Const("H1");
-    cout << "axiom := " << thm_v << endl;
-
-    theorem_rw trw(thm_t, thm_v);
     context ctx;
-    trw(ctx, y_plus_zero);
+    pair<expr, expr> result = add_comm_thm_rewriter(ctx, a_plus_b, Nat);
-    return 0;
+    expr concl = mk_eq(a_plus_b, result.first);
+    expr proof = result.second;
+
+    cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
+    cout << "         " << concl << " := " << proof << endl;
+
+    lean_assert(concl == mk_eq(a_plus_b, b_plus_a));
+    lean_assert(proof == mk_app(mk_app(Const("ADD_COMM"), a), b));
+    env.add_theorem("New_theorem1", concl, proof);
+}
+
+static void theorem_rewrite2_tst() {
+    cout << "=== theorem_rewrite2_tst() ===" << endl;
+    // Theorem:     Pi(x : N), x + 0 = x := ADD_ID x
+    // Term   :     a + 0
+    // Result :     (a, ADD_ID a)
+    expr a           = Const("a");                  // a    : at
+    expr zero        = nVal(0);                     // zero : Nat
+    expr a_plus_zero = nAdd(a, zero);
+    expr add_id_thm_type = Pi("x", Nat,
+                           Eq(nAdd(Const("x"), zero), Const("x")));
+    expr add_id_thm_body = Const("ADD_ID");
+
+    environment env = mk_toplevel();
+    env.add_var("a", Nat);
+    env.add_axiom("ADD_ID", add_id_thm_type); // ADD_ID : Pi (x : N), x = x + 0
+
+    // Rewriting
+    theorem_rewrite add_id_thm_rewriter(add_id_thm_type, add_id_thm_body);
+    context ctx;
+    pair<expr, expr> result = add_id_thm_rewriter(ctx, a_plus_zero, Nat);
+    expr concl = mk_eq(a_plus_zero, result.first);
+    expr proof = result.second;
+
+    cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
+    cout << "         " << concl << " := " << proof << endl;
+
+    lean_assert(concl == mk_eq(a_plus_zero, a));
+    lean_assert(proof == mk_app(Const("ADD_ID"), a));
+    env.add_theorem("New_theorem2", concl, proof);
+}
+
+static void then_rewrite1_tst() {
+    cout << "=== then_rewrite1_tst() ===" << endl;
+    // Theorem1:     Pi(x y : N), x + y = y + x := ADD_COMM x y
+    // Theorem2:     Pi(x : N)  , x + 0 = x     := ADD_ID x
+    // Term    :     0 + a
+    // Result :      (a, TRANS (ADD_COMM 0 a) (ADD_ID a))
+
+    expr a           = Const("a");                  // a  : Nat
+    expr zero        = nVal(0);                     // zero : Nat
+    expr a_plus_zero = nAdd(a, zero);
+    expr zero_plus_a = nAdd(zero, a);
+    expr add_comm_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+    expr add_comm_thm_body = Const("ADD_COMM");
+    expr add_id_thm_type = Pi("x", Nat,
+                           Eq(nAdd(Const("x"), zero), Const("x")));
+    expr add_id_thm_body = Const("ADD_ID");
+
+    environment env = mk_toplevel();
+    env.add_var("a", Nat);
+    env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
+    env.add_axiom("ADD_ID", add_id_thm_type); // ADD_ID : Pi (x : N), x = x + 0
+
+    // Rewriting
+    theorem_rewrite add_comm_thm_rewriter(add_comm_thm_type, add_comm_thm_body);
+    theorem_rewrite add_id_thm_rewriter(add_id_thm_type, add_id_thm_body);
+    then_rewrite    then_rewriter1(add_comm_thm_rewriter, add_id_thm_rewriter);
+    context ctx;
+    pair<expr, expr> result = then_rewriter1(ctx, zero_plus_a, Nat);
+    expr concl = mk_eq(zero_plus_a, result.first);
+    expr proof = result.second;
+
+    cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
+    cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
+    cout << "         " << concl << " := " << proof << endl;
+
+    lean_assert(concl == mk_eq(zero_plus_a, a));
+    lean_assert(proof == Trans(Nat, zero_plus_a, a_plus_zero, a,
+                               mk_app(mk_app(Const("ADD_COMM"), zero), a), mk_app(Const("ADD_ID"), a)));
+
+    env.add_theorem("New_theorem3", concl, proof);
+}
+
+static void then_rewrite2_tst() {
+    cout << "=== then_rewrite2_tst() ===" << endl;
+    // Theorem1:     Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
+    // Theorem2:     Pi(x y : N),  x + y       = y + x       := ADD_COMM x y
+    // Theorem3:     Pi(x : N),    x + 0       = x           := ADD_ID x
+    // Term    :     0 + (a + 0)
+    // Result :      (a, TRANS (ADD_ASSOC 0 a 0)         // (0 + a) + 0
+    //                         (ADD_ID (0 + a))          // 0 + a
+    //                         (ADD_COMM 0 a)            // a + 0
+    //                         (ADD_ID a))               // a
+
+    expr a           = Const("a");                  // a  : Nat
+    expr zero        = nVal(0);                     // zero : Nat
+    expr zero_plus_a  = nAdd(zero, a);
+    expr a_plus_zero  = nAdd(a, zero);
+    expr zero_plus_a_plus_zero  = nAdd(zero, nAdd(a, zero));
+    expr zero_plus_a_plus_zero_ = nAdd(nAdd(zero, a), zero);
+    expr add_assoc_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Pi("z", Nat,
+                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
+                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+    expr add_assoc_thm_body = Const("ADD_ASSOC");
+    expr add_comm_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+    expr add_comm_thm_body = Const("ADD_COMM");
+    expr add_id_thm_type = Pi("x", Nat,
+                           Eq(nAdd(Const("x"), zero), Const("x")));
+    expr add_id_thm_body = Const("ADD_ID");
+
+    environment env = mk_toplevel();
+    env.add_var("a", Nat);
+    env.add_axiom("ADD_ASSOC", add_assoc_thm_type); // ADD_ASSOC : Pi (x, y, z : N), x + (y + z) = (x + y) + z
+    env.add_axiom("ADD_COMM", add_comm_thm_type);   // ADD_COMM  : Pi (x, y: N), x + y = y + z
+    env.add_axiom("ADD_ID", add_id_thm_type);       // ADD_ID    : Pi (x : N), x = x + 0
+
+    // Rewriting
+    theorem_rewrite add_assoc_thm_rewriter(add_assoc_thm_type, add_assoc_thm_body);
+    theorem_rewrite add_comm_thm_rewriter(add_comm_thm_type, add_comm_thm_body);
+    theorem_rewrite add_id_thm_rewriter(add_id_thm_type, add_id_thm_body);
+    then_rewrite    then_rewriter2(then_rewrite(add_assoc_thm_rewriter, add_id_thm_rewriter),
+                                   then_rewrite(add_comm_thm_rewriter, add_id_thm_rewriter));
+    context ctx;
+    pair<expr, expr> result = then_rewriter2(ctx, zero_plus_a_plus_zero, Nat);
+    expr concl = mk_eq(zero_plus_a_plus_zero, result.first);
+    expr proof = result.second;
+    cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << endl;
+    cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
+    cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
+    cout << "         " << concl << " := " << proof << endl;
+
+    lean_assert(concl == mk_eq(zero_plus_a_plus_zero, a));
+    lean_assert(proof == Trans(Nat, zero_plus_a_plus_zero, zero_plus_a, a,
+                               Trans(Nat, zero_plus_a_plus_zero, zero_plus_a_plus_zero_, zero_plus_a,
+                                     mk_app(mk_app(mk_app(Const("ADD_ASSOC"), zero), a), zero),
+                                     mk_app(Const("ADD_ID"), zero_plus_a)),
+                               Trans(Nat, zero_plus_a, a_plus_zero, a,
+                                     mk_app(mk_app(Const("ADD_COMM"), zero), a),
+                                     mk_app(Const("ADD_ID"), a))));
+
+    env.add_theorem("New_theorem4", concl, proof);
+}
+
+
+static void orelse_rewrite1_tst() {
+    cout << "=== orelse_rewrite1_tst() ===" << endl;
+    // Theorem1:     Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
+    // Theorem2:     Pi(x y : N),  x + y       = y + x       := ADD_COMM x y
+    // Term   :     a + b
+    // Result :     (b + a, ADD_COMM a b)
+    expr a        = Const("a");                  // a  : Nat
+    expr b        = Const("b");                  // b  : Nat
+    expr a_plus_b = nAdd(a, b);
+    expr b_plus_a = nAdd(b, a);
+    expr add_assoc_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Pi("z", Nat,
+                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
+                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+    expr add_assoc_thm_body = Const("ADD_ASSOC");
+    expr add_comm_thm_type = Pi("x", Nat,
+                                Pi("y", Nat,
+                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+    expr add_comm_thm_body = Const("ADD_COMM");
+
+    environment env = mk_toplevel();
+    env.add_var("a", Nat);
+    env.add_var("b", Nat);
+    env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
+
+    // Rewriting
+    theorem_rewrite add_assoc_thm_rewriter(add_assoc_thm_type, add_assoc_thm_body);
+    theorem_rewrite add_comm_thm_rewriter(add_comm_thm_type, add_comm_thm_body);
+    orelse_rewrite  add_assoc_or_comm_thm_rewriter(add_assoc_thm_rewriter, add_comm_thm_rewriter);
+    context ctx;
+    pair<expr, expr> result = add_assoc_or_comm_thm_rewriter(ctx, a_plus_b, Nat);
+    expr concl = mk_eq(a_plus_b, result.first);
+    expr proof = result.second;
+
+    cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << endl;
+    cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
+    cout << "         " << concl << " := " << proof << endl;
+
+    lean_assert(concl == mk_eq(a_plus_b, b_plus_a));
+    lean_assert(proof == mk_app(mk_app(Const("ADD_COMM"), a), b));
+    env.add_theorem("New_theorem5", concl, proof);
+}
+
+int main() {
+    theorem_rewrite1_tst();
+    theorem_rewrite2_tst();
+    then_rewrite1_tst();
+    then_rewrite2_tst();
+    orelse_rewrite1_tst();
+    return has_violations() ? 1 : 0;
 }