michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Add then and orelse rewrite combinators and tests

Browse source
This commit is contained in:
Soonho Kong 2013-09-23 18:53:39 -07:00
parent 9ba6068858
commit 81c9de229b
5 changed files with 366 additions and 143 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

106
src/library/rewrite/fo_match.cpp
View file

@ -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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_var : (" << 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));
cout << "match_var MATCHED : " << s << endl;
s.insert(std::make_pair(idx, t));
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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_constant : (" << 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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_value : (" << 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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_app : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
if (!is_app(t))
return false;
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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
cout << "fun (" << abst_name(p)
<< " : " << abst_domain(p)
<< "), " << abst_body(p) << endl;
lean_trace("fo_match", tout << "match_lambda : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
lean_trace("fo_match", tout << "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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
cout << "Pi (" << abst_name(p)
<< " : " << abst_domain(p)
<< "), " << abst_body(p) << endl;
lean_trace("fo_match", tout << "match_pi : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
lean_trace("fo_match", tout << "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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_type : (" << 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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_eq : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
if (!is_eq(t))
return false;
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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_let : (" << 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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match_meta : (" << 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 : ("
<< p << ", "
<< t << ", "
<< o << ", "
<< s << ")"
<< endl;
lean_trace("fo_match", tout << "match : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;);
switch (p.kind()) {
case expr_kind::Var:
return match_var(p, t, o, s);

2
src/library/rewrite/fo_match.h
View file

@ -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:

99
src/library/rewrite/rewrite.cpp
View file

@ -4,10 +4,13 @@ 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/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)
: thm_t(t), thm_v(v), num_args(0) {
cout << "================= Theorem Rewrite Constructor Start ===================" << endl;
cout << "Type = " << thm_t << endl;
cout << "Body = " << thm_v << endl;
theorem_rewrite::theorem_rewrite(expr const & type, expr const & body)
: thm_type(type), thm_body(body), num_args(0) {
lean_trace("rewrite", tout << "Type = " << thm_type << endl;);
lean_trace("rewrite", tout << "Body = " << thm_body << endl;);
// We expect the theorem is in the form of
// Pi (x_1 : t_1 ... x_n : t_n), t = s
expr tmp = t;
while (is_pi(tmp)) {
tmp = abst_body(tmp);
// Pi (x_1 : t_1 ... x_n : t_n), pattern = rhs
pattern = type;
while (is_pi(pattern)) {
pattern = abst_body(pattern);
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;
if (!is_eq(pattern)) {
lean_trace("rewrite", tout << "Theorem " << thm_type << " is not in the form of "
<< "Pi (x_1 : t_1 ... x_n : t_n), pattern = rhs" << endl;);
}
cout << "OK. Number of Arg = " << num_args << endl;
cout << "================= Theorem Rewrite Constructor END ===================" << endl;
rhs = eq_rhs(pattern);
pattern = eq_lhs(pattern);
lean_trace("rewrite", tout << "Number of Arg = " << num_args << endl;);
}
void theorem_rw::operator()(context & ctx, expr const & t) {
cout << "================= Theorem Rewrite () START ===================" << endl;
cout << "Context = " << ctx << endl;
cout << "Term = " << t << endl;
expr tmp = thm_t;
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);
pair<expr, expr> theorem_rewrite::operator()(context & ctx, expr const & v, expr const & ) const throw(rewrite_exception) {
lean_trace("rewrite", tout << "Context = " << ctx << endl;);
lean_trace("rewrite", tout << "Term = " << v << endl;);
lean_trace("rewrite", tout << "Pattern = " << pattern << endl;);
lean_trace("rewrite", tout << "Num Args = " << num_args << endl;);
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));
}
}

49
src/library/rewrite/rewrite.h
View file

@ -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_v;
expr const & thm_type;
expr const & thm_body;
expr pattern;
expr rhs;
unsigned num_args;
public:
theorem_rw(expr const & t, expr const & v);
void operator()(context & ctx, expr const & t);
theorem_rewrite(expr const & type, expr const & body);
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();
}
};
}

247
src/tests/library/rewrite/rewrite.cpp
View file

@ -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);
expr x = Const("x"); // x : Nat
expr y = Const("y"); // y : Nat
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);
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
expr thm_t = Pi("x", Nat, Eq(nAdd(Const("x"), nVal(0)), Const("x"))); // Pi (x : Z), x + 0 = x
cout << "theorem := " << thm_t << endl;
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);
// Rewriting
theorem_rewrite add_comm_thm_rewriter(add_comm_thm_type, add_comm_thm_body);
context ctx;
trw(ctx, y_plus_zero);
return 0;
pair<expr, expr> result = add_comm_thm_rewriter(ctx, a_plus_b, Nat);
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;
}