test(library/rewriter): add more tests
This commit is contained in:
Soonho Kong
parent
7c0b56ad0d
commit
c50bc13be0
1 changed files with 382 additions and 32 deletions
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@ -15,14 +15,15 @@ Author: Soonho Kong
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#include "library/rewriter/rewriter.h"
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#include "library/basic_thms.h"
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#include "library/printer.h"
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#include "library/kernel_exception_formatter.h"
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using namespace lean;
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using std::cout;
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using std::pair;
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using std::endl;
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using lean::endl;
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static void theorem_rewriter1_tst() {
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cout << "=== theorem_rewriter1_tst() ===" << endl;
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cout << "=== theorem_rewriter1_tst() ===" << std::endl;
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// Theorem: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Term : a + b
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// Result : (b + a, ADD_COMM a b)
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@ -47,16 +48,16 @@ static void theorem_rewriter1_tst() {
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expr concl = mk_eq(a_plus_b, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
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cout << " " << concl << " := " << proof << endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert(concl == mk_eq(a_plus_b, b_plus_a));
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lean_assert(proof == Const("ADD_COMM")(a, b));
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lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
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lean_assert_eq(proof, Const("ADD_COMM")(a, b));
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env.add_theorem("New_theorem1", concl, proof);
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}
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static void theorem_rewriter2_tst() {
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cout << "=== theorem_rewriter2_tst() ===" << endl;
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cout << "=== theorem_rewriter2_tst() ===" << std::endl;
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// Theorem: Pi(x : N), x + 0 = x := ADD_ID x
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// Term : a + 0
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// Result : (a, ADD_ID a)
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@ -78,16 +79,16 @@ static void theorem_rewriter2_tst() {
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expr concl = mk_eq(a_plus_zero, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
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cout << " " << concl << " := " << proof << endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert(concl == mk_eq(a_plus_zero, a));
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lean_assert(proof == Const("ADD_ID")(a));
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lean_assert_eq(concl, mk_eq(a_plus_zero, a));
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lean_assert_eq(proof, Const("ADD_ID")(a));
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env.add_theorem("New_theorem2", concl, proof);
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}
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static void then_rewriter1_tst() {
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cout << "=== then_rewriter1_tst() ===" << endl;
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cout << "=== then_rewriter1_tst() ===" << std::endl;
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// Theorem1: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Theorem2: Pi(x : N) , x + 0 = x := ADD_ID x
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// Term : 0 + a
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@ -119,11 +120,11 @@ static void then_rewriter1_tst() {
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expr concl = mk_eq(zero_plus_a, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
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cout << " " << concl << " := " << proof << endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert(concl == mk_eq(zero_plus_a, a));
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lean_assert_eq(concl, mk_eq(zero_plus_a, a));
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lean_assert(proof == Trans(Nat, zero_plus_a, a_plus_zero, a,
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Const("ADD_COMM")(zero, a), Const("ADD_ID")(a)));
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@ -131,7 +132,7 @@ static void then_rewriter1_tst() {
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}
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static void then_rewriter2_tst() {
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cout << "=== then_rewriter2_tst() ===" << endl;
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cout << "=== then_rewriter2_tst() ===" << std::endl;
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// Theorem1: Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
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// Theorem2: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Theorem3: Pi(x : N), x + 0 = x := ADD_ID x
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@ -179,12 +180,12 @@ static void then_rewriter2_tst() {
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pair<expr, expr> result = then_rewriter2(env, ctx, zero_plus_a_plus_zero);
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expr concl = mk_eq(zero_plus_a_plus_zero, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
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cout << " " << concl << " := " << proof << endl;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert(concl == mk_eq(zero_plus_a_plus_zero, a));
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lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
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lean_assert(proof == Trans(Nat, zero_plus_a_plus_zero, a_plus_zero, a,
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Trans(Nat, zero_plus_a_plus_zero, zero_plus_a, a_plus_zero,
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Trans(Nat, zero_plus_a_plus_zero, zero_plus_a_plus_zero_, zero_plus_a,
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@ -195,9 +196,8 @@ static void then_rewriter2_tst() {
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env.add_theorem("New_theorem4", concl, proof);
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}
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static void orelse_rewriter1_tst() {
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cout << "=== orelse_rewriter1_tst() ===" << endl;
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cout << "=== orelse_rewriter1_tst() ===" << std::endl;
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// Theorem1: Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
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// Theorem2: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Term : a + b
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@ -238,21 +238,371 @@ static void orelse_rewriter1_tst() {
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expr concl = mk_eq(a_plus_b, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << endl;
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cout << " " << concl << " := " << proof << endl;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert(concl == mk_eq(a_plus_b, b_plus_a));
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lean_assert(proof == Const("ADD_COMM")(a, b));
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lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
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lean_assert_eq(proof, Const("ADD_COMM")(a, b));
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env.add_theorem("New_theorem5", concl, proof);
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}
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static void orelse_rewriter2_tst() {
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cout << "=== orelse_rewriter2_tst() ===" << std::endl;
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// Theorem1: Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
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// Term : a + b
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// Result : Fail
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expr a = Const("a"); // a : Nat
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expr b = Const("b"); // b : Nat
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expr zero = nVal(0); // zero : Nat
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expr a_plus_b = nAdd(a, b);
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expr b_plus_a = nAdd(b, a);
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expr add_assoc_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Pi("z", Nat,
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Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
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nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
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expr add_assoc_thm_body = Const("ADD_ASSOC");
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expr add_id_thm_type = Pi("x", Nat,
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Eq(nAdd(Const("x"), zero), Const("x")));
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expr add_id_thm_body = Const("ADD_ID");
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environment env = mk_toplevel();
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env.add_var("a", Nat);
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env.add_var("b", Nat);
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env.add_axiom("ADD_ASSOC", add_assoc_thm_type);
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env.add_axiom("ADD_ID", add_id_thm_type);
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// Rewriting
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rewriter add_assoc_thm_rewriter = mk_theorem_rewriter(add_assoc_thm_type, add_assoc_thm_body);
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rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
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rewriter add_orelse_rewriter = mk_orelse_rewriter(add_assoc_thm_rewriter, add_id_thm_rewriter);
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context ctx;
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try {
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pair<expr, expr> result = add_orelse_rewriter(env, ctx, a_plus_b);
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lean_unreachable();
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} catch (rewriter_exception & ) {
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// Do nothing
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cout << "Exception Caught!" << std::endl;
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return;
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}
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lean_unreachable();
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}
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static void try_rewriter1_tst() {
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cout << "=== try_rewriter1_tst() ===" << std::endl;
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// Theorem1: Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
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// Theorem2: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Theorem3: Pi (x : N), x = x + 0 := ADD_ID x
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// Term : a + b
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// Result : (b + a, ADD_COMM a b)
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expr a = Const("a"); // a : Nat
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expr b = Const("b"); // b : Nat
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expr zero = nVal(0); // zero : Nat
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expr a_plus_b = nAdd(a, b);
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expr b_plus_a = nAdd(b, a);
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expr add_assoc_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Pi("z", Nat,
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Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
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nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
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expr add_assoc_thm_body = Const("ADD_ASSOC");
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expr add_comm_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
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expr add_comm_thm_body = Const("ADD_COMM");
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expr add_id_thm_type = Pi("x", Nat,
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Eq(nAdd(Const("x"), zero), Const("x")));
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expr add_id_thm_body = Const("ADD_ID");
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environment env = mk_toplevel();
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env.add_var("a", Nat);
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env.add_var("b", Nat);
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env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
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// Rewriting
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rewriter add_assoc_thm_rewriter = mk_theorem_rewriter(add_assoc_thm_type, add_assoc_thm_body);
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rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
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rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
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rewriter add_try_rewriter = mk_try_rewriter({add_assoc_thm_rewriter,
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add_id_thm_rewriter});
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context ctx;
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pair<expr, expr> result = add_try_rewriter(env, ctx, a_plus_b);
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expr concl = mk_eq(a_plus_b, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert_eq(concl, mk_eq(a_plus_b, a_plus_b));
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lean_assert_eq(proof, Const("Refl")(Nat, a_plus_b));
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env.add_theorem("New_theorem6", concl, proof);
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}
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static void try_rewriter2_tst() {
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cout << "=== try_rewriter2_tst() ===" << std::endl;
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// Theorem1: Pi(x y z: N), x + (y + z) = (x + y) + z := ADD_ASSOC x y z
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// Theorem2: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Term : a + b
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// Result : (b + a, ADD_COMM a b)
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expr a = Const("a"); // a : Nat
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expr b = Const("b"); // b : Nat
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expr zero = nVal(0); // zero : Nat
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expr a_plus_b = nAdd(a, b);
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expr b_plus_a = nAdd(b, a);
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expr add_assoc_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Pi("z", Nat,
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Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
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nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
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expr add_assoc_thm_body = Const("ADD_ASSOC");
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expr add_comm_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
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expr add_comm_thm_body = Const("ADD_COMM");
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expr add_id_thm_type = Pi("x", Nat,
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Eq(nAdd(Const("x"), zero), Const("x")));
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expr add_id_thm_body = Const("ADD_ID");
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environment env = mk_toplevel();
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env.add_var("a", Nat);
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env.add_var("b", Nat);
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env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
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// Rewriting
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rewriter add_assoc_thm_rewriter = mk_theorem_rewriter(add_assoc_thm_type, add_assoc_thm_body);
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rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
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rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
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rewriter add_try_rewriter = mk_try_rewriter({add_assoc_thm_rewriter,
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add_comm_thm_rewriter,
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add_id_thm_rewriter});
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context ctx;
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pair<expr, expr> result = add_try_rewriter(env, ctx, a_plus_b);
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expr concl = mk_eq(a_plus_b, result.first);
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expr proof = result.second;
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cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
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cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
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cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
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cout << " " << concl << " := " << proof << std::endl;
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lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
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lean_assert_eq(proof, Const("ADD_COMM")(a, b));
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env.add_theorem("try2", concl, proof);
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}
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static void app_rewriter1_tst() {
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cout << "=== app_rewriter1_tst() ===" << std::endl;
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// Theorem: Pi(x y : N), x + y = y + x := ADD_COMM x y
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// Term : f (a + b)
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// Result : (f (b + a), ADD_COMM a b)
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expr a = Const("a"); // a : Nat
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expr b = Const("b"); // b : Nat
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expr f1 = Const("f1"); // f : Nat -> Nat
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expr f2 = Const("f2"); // f : Nat -> Nat -> Nat
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expr f3 = Const("f3"); // f : Nat -> Nat -> Nat -> Nat
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expr f4 = Const("f4"); // f : Nat -> Nat -> Nat -> Nat -> Nat
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expr zero = nVal(0); // zero : Nat
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expr a_plus_b = nAdd(a, b);
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expr b_plus_a = nAdd(b, a);
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expr add_comm_thm_type = Pi("x", Nat,
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Pi("y", Nat,
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Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
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expr add_comm_thm_body = Const("ADD_COMM");
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environment env = mk_toplevel();
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env.add_var("f1", Nat >> Nat);
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env.add_var("f2", Nat >> (Nat >> Nat));
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env.add_var("f3", Nat >> (Nat >> (Nat >> Nat)));
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env.add_var("f4", Nat >> (Nat >> (Nat >> (Nat >> Nat))));
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env.add_var("a", Nat);
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env.add_var("b", Nat);
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env.add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
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// Rewriting
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rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
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rewriter add_try_comm_rewriter = mk_try_rewriter(add_comm_thm_rewriter);
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rewriter app_try_comm_rewriter = mk_app_rewriter(add_try_comm_rewriter);
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context ctx;
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cout << "RW = " << app_try_comm_rewriter << std::endl;
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expr v = f1(nVal(0));
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pair<expr, expr> result = app_try_comm_rewriter(env, ctx, v);
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expr concl = mk_eq(v, result.first);
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expr proof = result.second;
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cout << "Concl = " << concl << std::endl
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<< "Proof = " << proof << std::endl;
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lean_assert_eq(concl, mk_eq(v, f1(nVal(0))));
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lean_assert_eq(proof, Const("Refl")(Nat >> Nat, f1));
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env.add_theorem("app_rewriter1", concl, proof);
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// ==========================================================
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cout << "====================================================" << std::endl;
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v = f1(a_plus_b);
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result = app_try_comm_rewriter(env, ctx, v);
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concl = mk_eq(v, result.first);
|
||||
proof = result.second;
|
||||
cout << "Concl = " << concl << std::endl
|
||||
<< "Proof = " << proof << std::endl;
|
||||
lean_assert_eq(concl, mk_eq(v, f1(b_plus_a)));
|
||||
lean_assert_eq(proof,
|
||||
Const("Congr2")(Nat, Fun(name("_"), Nat, Nat), a_plus_b, b_plus_a, f1, Const("ADD_COMM")(a, b)));
|
||||
env.add_theorem("app_rewriter2", concl, proof);
|
||||
// ==========================================================
|
||||
cout << "====================================================" << std::endl;
|
||||
v = f4(nVal(0), a_plus_b, nVal(0), b_plus_a);
|
||||
result = app_try_comm_rewriter(env, ctx, v);
|
||||
concl = mk_eq(v, result.first);
|
||||
proof = result.second;
|
||||
cout << "Concl = " << concl << std::endl
|
||||
<< "Proof = " << proof << std::endl;
|
||||
lean_assert_eq(concl, mk_eq(v, f4(nVal(0), b_plus_a, nVal(0), a_plus_b)));
|
||||
// Congr Nat (fun _ : Nat, Nat) (f4 0 (Nat::add a b) 0) (f4 0 (Nat::add b a) 0) (Nat::add b a) (Nat::add a b) (Congr1 Nat (fun _ : Nat, (Nat -> Nat)) (f4 0 (Nat::add a b)) (f4 0 (Nat::add b a)) 0 (Congr2 Nat (fun _ : Nat, (Nat -> Nat -> Nat)) (Nat::add a b) (Nat::add b a) (f4 0) (ADD_COMM a b))) (ADD_COMM b a)
|
||||
|
||||
lean_assert_eq(proof,
|
||||
Const("Congr")(Nat, Fun(name("_"), Nat, Nat), f4(zero, a_plus_b, zero), f4(zero, b_plus_a, zero),
|
||||
b_plus_a, a_plus_b,
|
||||
Const("Congr1")(Nat, Fun(name("_"), Nat, Nat >> Nat), f4(zero, a_plus_b),
|
||||
f4(zero, b_plus_a), zero,
|
||||
Const("Congr2")(Nat, Fun(name("_"), Nat, Nat >> (Nat >> Nat)),
|
||||
a_plus_b, b_plus_a, f4(zero),
|
||||
Const("ADD_COMM")(a, b))),
|
||||
Const("ADD_COMM")(b, a)));
|
||||
env.add_theorem("app_rewriter3", concl, proof);
|
||||
}
|
||||
|
||||
static void repeat_rewriter1_tst() {
|
||||
cout << "=== repeat_rewriter1_tst() ===" << std::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
|
||||
rewriter add_assoc_thm_rewriter = mk_theorem_rewriter(add_assoc_thm_type, add_assoc_thm_body);
|
||||
rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
|
||||
rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
|
||||
rewriter or_rewriter = mk_orelse_rewriter({add_assoc_thm_rewriter,
|
||||
add_id_thm_rewriter,
|
||||
add_comm_thm_rewriter});
|
||||
rewriter repeat_rw = mk_repeat_rewriter(or_rewriter);
|
||||
context ctx;
|
||||
pair<expr, expr> result = repeat_rw(env, ctx, zero_plus_a_plus_zero);
|
||||
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 << std::endl;
|
||||
cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
|
||||
cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
|
||||
cout << " " << concl << " := " << proof << std::endl;
|
||||
|
||||
lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
|
||||
env.add_theorem("repeat_thm1", concl, proof);
|
||||
}
|
||||
|
||||
static void repeat_rewriter2_tst() {
|
||||
cout << "=== repeat_rewriter2_tst() ===" << std::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
|
||||
rewriter add_assoc_thm_rewriter = mk_theorem_rewriter(add_assoc_thm_type, add_assoc_thm_body);
|
||||
rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
|
||||
rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
|
||||
rewriter or_rewriter = mk_orelse_rewriter({add_assoc_thm_rewriter,
|
||||
add_id_thm_rewriter,
|
||||
add_comm_thm_rewriter});
|
||||
rewriter try_rw = mk_try_rewriter(or_rewriter);
|
||||
rewriter repeat_rw = mk_repeat_rewriter(try_rw);
|
||||
context ctx;
|
||||
pair<expr, expr> result = repeat_rw(env, ctx, zero_plus_a_plus_zero);
|
||||
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 << std::endl;
|
||||
cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
|
||||
cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
|
||||
cout << " " << concl << " := " << proof << std::endl;
|
||||
|
||||
lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
|
||||
env.add_theorem("repeat_thm2", concl, proof);
|
||||
}
|
||||
|
||||
int main() {
|
||||
theorem_rewriter1_tst();
|
||||
theorem_rewriter2_tst();
|
||||
then_rewriter1_tst();
|
||||
then_rewriter2_tst();
|
||||
orelse_rewriter1_tst();
|
||||
orelse_rewriter2_tst();
|
||||
try_rewriter1_tst();
|
||||
try_rewriter2_tst();
|
||||
app_rewriter1_tst();
|
||||
repeat_rewriter1_tst();
|
||||
repeat_rewriter2_tst();
|
||||
return has_violations() ? 1 : 0;
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in a new issue