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feat(library/basic_thms): add ExistsElim theorem

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-12-16 12:43:34 -08:00
parent 8f5c2b7d9f
commit de53e92de8
3 changed files with 15 additions and 11 deletions
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1
src/frontends/lean/notation.cpp
View file

@ -121,6 +121,7 @@ void init_builtin_notation(frontend & f) {
f.mark_implicit_arguments(mk_congr_fn(), 6);
f.mark_implicit_arguments(mk_forall_elim_fn(), 2);
f.mark_implicit_arguments(mk_forall_intro_fn(), 2);
f.mark_implicit_arguments(mk_exists_elim_fn(), 3);
f.mark_implicit_arguments(mk_exists_intro_fn(), 2);
}
}

10
src/library/basic_thms.cpp
View file

@ -43,6 +43,7 @@ MK_CONSTANT(eqt_elim_fn, name("EqTElim"));
MK_CONSTANT(eqt_intro_fn, name("EqTIntro"));
MK_CONSTANT(forall_elim_fn, name("ForallElim"));
MK_CONSTANT(forall_intro_fn, name("ForallIntro"));
MK_CONSTANT(exists_elim_fn, name("ExistsElim"));
MK_CONSTANT(exists_intro_fn, name("ExistsIntro"));
#if 0
@ -74,6 +75,7 @@ void import_basic_thms(environment const & env) {
expr A1 = Const("A1");
expr B1 = Const("B1");
expr a1 = Const("a1");
expr R = Const("R");
expr A_pred = A >> Bool;
expr q_type = Pi({A, TypeU}, A_pred >> Bool);
@ -286,6 +288,14 @@ void import_basic_thms(environment const & env) {
Symm(A_pred, Fun({x, A}, P(x)), P, Eta(A, Fun({x, A}, Bool), P)), // P == fun x : A, P x
Abst(A, Fun({x, A}, Bool), Fun({x, A}, P(x)), Fun({x, A}, True), Fun({x, A}, EqTIntro(P(x), H(x)))))));
// ExistsElim : Pi (A : Type U) (P : A -> Bool) (B : Bool) (H1 : exists x : A, P x) (H2 : Pi (a : A) (H : P a), B) : B :=
env->add_theorem(exists_elim_fn_name, Pi({{A, TypeU}, {P, A_pred}, {B, Bool}, {H1, mk_exists(A, P)}, {H2, Pi({{a, A}, {H, P(a)}}, B)}}, B),
Fun({{A, TypeU}, {P, A_pred}, {B, Bool}, {H1, mk_exists(A, P)}, {H2, Pi({{a, A}, {H, P(a)}}, B)}},
Refute(B, Fun({R, Not(B)},
Absurd(mk_forall(A, Fun({x, A}, Not(P(x)))),
ForallIntro(A, Fun({x, A}, Not(P(x))), Fun({a, A}, MT(P(a), B, Discharge(P(a), B, Fun({H, P(a)}, H2(a, H))), R))),
H1)))));
// ExistsIntro : Pi (A : Type u) (P : A -> bool) (a : A) (H : P a), exists A P
env->add_theorem(exists_intro_fn_name, Pi({{A, TypeU}, {P, A_pred}, {a, A}, {H, P(a)}}, mk_exists(A, P)),
Fun({{A, TypeU}, {P, A_pred}, {a, A}, {H, P(a)}},

15
src/library/basic_thms.h
View file

@ -130,21 +130,14 @@ expr mk_forall_intro_fn();
// \brief (Theorem) {A : Type u} {P : A -> bool} (H : Pi (x : A), P x) |- ForallIntro(A, P, H) : forall x : A, P
inline expr ForallIntro(expr const & A, expr const & P, expr const & H) { return mk_app(mk_forall_intro_fn(), A, P, H); }
expr mk_exists_elim_fn();
// \brief (Theorem) {A : Type U} {P : A -> Bool} {B : Bool} (H1 : exists x : A, P x) (H2 : Pi (a : A) (H : P a) |- ExistsElim(A, P, B, H1, H2) : B
inline expr ExistsElim(expr const & A, expr const & P, expr const & B, expr const & H1, expr const & H2) { return mk_app({mk_exists_elim_fn(), A, P, B, H1, H2}); }
expr mk_exists_intro_fn();
// \brief (Theorem) {A : Type u}, {P : A -> Bool}, a : A, H : P a |- ExistsIntro(A, P, a, H) : exists x : A, P
inline expr ExistsIntro(expr const & A, expr const & P, expr const & a, expr const & H) { return mk_app(mk_exists_intro_fn(), A, P, a, H); }
/** \brief Add basic theorems to Environment */
void import_basic_thms(environment const & env);
#if 0
expr mk_ext_fn();
bool is_ext_fn(expr const & e);
expr mk_foralli_fn();
bool is_foralli_fn(expr const & e);
expr mk_domain_inj_fn();
bool is_domain_inj_fn(expr const & e);
expr mk_range_inj_fn();
bool is_range_inj_fn(expr const & e);
#endif
}