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Add subst proof rule. Define symm and trans using subst.

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-08-06 02:03:22 -07:00
parent 30513398bb
commit 33d2dd2d8b
5 changed files with 43 additions and 6 deletions
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19
src/kernel/builtin.cpp
View file

@ -148,6 +148,7 @@ MK_CONSTANT(forall_fn, name("forall"));
MK_CONSTANT(exists_fn, name("exists"));
MK_CONSTANT(refl_fn, name("refl"));
MK_CONSTANT(subst_fn, name("subst"));
MK_CONSTANT(symm_fn, name("symm"));
MK_CONSTANT(trans_fn, name("trans"));
MK_CONSTANT(congr_fn, name("congr"));
@ -157,6 +158,8 @@ MK_CONSTANT(foralli_fn, name("foralli"));
MK_CONSTANT(domain_inj_fn, name("domain_inj"));
MK_CONSTANT(range_inj_fn, name("range_inj"));
void add_basic_theory(environment & env) {
env.define_uvar(uvar_name(m_lvl), level() + LEAN_DEFAULT_LEVEL_SEPARATION);
env.define_uvar(uvar_name(u_lvl), m_lvl + LEAN_DEFAULT_LEVEL_SEPARATION);
@ -195,10 +198,18 @@ void add_basic_theory(environment & env) {
// refl : Pi (A : Type u) (a : A), a = a
env.add_axiom(refl_fn_name, Fun(A, u_type(), Fun(a, A, eq(a, a))));
// symm : Pi (A : Type u) (a b : A) (H : a = b), b = a
env.add_axiom(symm_fn_name, Fun(A, u_type(), Fun(a, A, Fun(b, A, Fun(H, eq(a, b), eq(b, a))))));
// trans: Pi (A: Type u) (a b c : A) (H1 : a = b) (H2 : b = c), a = c
env.add_axiom(trans_fn_name, Fun(A, u_type(), Fun(a, A, Fun(b, A, Fun(c, A, Fun(H1, eq(a, b), Fun(H2, eq(b, c), eq(a, c))))))));
// subst : Pi (A : Type u) (P : A -> bool) (a b : A) (H1 : P a) (H2 : a = b), P b
env.add_axiom(subst_fn_name, Fun(A, u_type(), Fun(P, A_pred, Fun(a, A, Fun(b, A, Fun(H1, P(a), Fun(H2, eq(a, b), P(b))))))));
// symm : Pi (A : Type u) (a b : A) (H : a = b), b = a :=
// subst A (fun x : A => x = a) a b (refl A a) H
env.add_definition(symm_fn_name, Fun(A, u_type(), Fun(a, A, Fun(b, A, Fun(H, eq(a, b), eq(b, a))))),
fun(A, u_type(), fun(a, A, fun(b, A, fun(H, eq(a, b), app(subst_fn(), A, fun(x, A, eq(x,a)), a, b, app(refl_fn(), A, a), H))))));
// trans: Pi (A: Type u) (a b c : A) (H1 : a = b) (H2 : b = c), a = c :=
// subst A (fun x : A => a = x) b c H1 H2
env.add_definition(trans_fn_name, Fun(A, u_type(), Fun(a, A, Fun(b, A, Fun(c, A, Fun(H1, eq(a, b), Fun(H2, eq(b, c), eq(a, c))))))),
fun(A, u_type(), fun(a, A, fun(b, A, fun(c, A, fun(H1, eq(a, b), fun(H2, eq(b, c), app(subst_fn(), A, fun(x, A, eq(a, x)), b, c, H1, H2))))))));
// congr : Pi (A : Type u) (B : A -> Type u) (f g : Pi (x : A) B x) (a b : A) (H1 : f = g) (H2 : a = b), f a = g b
expr piABx = Fun(x, A, B(x));
expr A_arrow_u = arrow(A, u_type());

2
src/kernel/builtin.h
View file

@ -92,6 +92,8 @@ inline expr mk_exists(expr const & A, expr const & P) { return app(exists_fn(),
expr refl_fn();
bool is_refl_fn(expr const & e);
expr subst_fn();
bool is_subst_fn(expr const & e);
expr symm_fn();
bool is_symm_fn(expr const & e);
expr trans_fn();

12
src/kernel/environment.cpp
View file

@ -28,8 +28,13 @@ environment::definition::definition(name const & n, expr const & t, expr const &
environment::definition::~definition() {
}
void environment::definition::display_header(std::ostream & out) const {
out << "Definition";
}
void environment::definition::display(std::ostream & out) const {
out << "Definition " << m_name << " : " << m_type << " := " << m_value << "\n";
display_header(out);
out << " " << m_name << " : " << m_type << " := " << m_value << "\n";
}
environment::fact::fact(name const & n, expr const & t):
@ -199,6 +204,11 @@ struct environment::imp {
m_object_dictionary.insert(std::make_pair(n, m_objects.back()));
}
void add_theorem(name const & n, expr const & t, expr const & v) {
m_objects.push_back(new theorem(n, t, v));
m_object_dictionary.insert(std::make_pair(n, m_objects.back()));
}
void add_axiom(name const & n, expr const & t) {
m_objects.push_back(new axiom(n, t));
m_object_dictionary.insert(std::make_pair(n, m_objects.back()));

10
src/kernel/environment.h
View file

@ -69,7 +69,7 @@ public:
*/
environment parent() const;
enum class object_kind { Definition, Var, Axiom };
enum class object_kind { Definition, Theorem, Var, Axiom };
/**
\brief Base class for environment objects
@ -92,6 +92,7 @@ public:
expr m_type;
expr m_value;
bool m_opaque;
virtual void display_header(std::ostream & out) const;
public:
definition(name const & n, expr const & t, expr const & v, bool opaque);
virtual ~definition();
@ -103,6 +104,13 @@ public:
virtual void display(std::ostream & out) const;
};
class theorem : public definition {
virtual void display_header(std::ostream & out) const { out << "Theorem"; }
public:
theorem(name const & n, expr const & t, expr const & v):definition(n, t, v, true) {}
virtual object_kind kind() const { return object_kind::Theorem; }
};
class fact : public object {
protected:
name m_name;

6
src/kernel/expr.h
View file

@ -260,6 +260,12 @@ inline expr app(expr const & e1, expr const & e2) { expr args[2] = {e1, e2}; ret
inline expr app(expr const & e1, expr const & e2, expr const & e3) { expr args[3] = {e1, e2, e3}; return app(3, args); }
inline expr app(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { expr args[4] = {e1, e2, e3, e4}; return app(4, args); }
inline expr app(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) { expr args[5] = {e1, e2, e3, e4, e5}; return app(5, args); }
inline expr app(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) {
expr args[6] = {e1, e2, e3, e4, e5, e6}; return app(6, args);
}
inline expr app(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6, expr const & e7) {
expr args[7] = {e1, e2, e3, e4, e5, e6, e7}; return app(7, args);
}
inline expr eq(expr const & l, expr const & r) { return expr(new expr_eq(l, r)); }
inline expr lambda(name const & n, expr const & t, expr const & e) { return expr(new expr_lambda(n, t, e)); }
inline expr lambda(char const * n, expr const & t, expr const & e) { return lambda(name(n), t, e); }