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Add Eta axiom

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-08-06 14:47:13 -07:00
parent 68d092f5ef
commit 3ff3eb6444
2 changed files with 14 additions and 5 deletions
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14
src/kernel/builtin.cpp
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@ -149,6 +149,7 @@ MK_CONSTANT(exists_fn, name("exists"));
MK_CONSTANT(refl_fn, name("refl"));
MK_CONSTANT(subst_fn, name("subst"));
MK_CONSTANT(eta_fn, name("eta"));
MK_CONSTANT(symm_fn, name("symm"));
MK_CONSTANT(trans_fn, name("trans"));
MK_CONSTANT(xtrans_fn, name("xtrans"));
@ -188,6 +189,11 @@ void add_basic_theory(environment & env) {
expr B1 = Const("B1");
expr a1 = Const("a1");
expr A_pred = A >> Bool;
expr q_type = Pi({A, TypeU}, A_pred >> Bool);
expr piABx = Pi({x, A}, B(x));
expr A_arrow_u = A >> TypeU;
// and(x, y) = (if bool x y false)
env.add_definition(and_fn_name, p2, Fun({{x, Bool}, {y, Bool}}, bIf(x, y, True)));
// or(x, y) = (if bool x true y)
@ -196,8 +202,6 @@ void add_basic_theory(environment & env) {
env.add_definition(not_fn_name, p1, Fun({x, Bool}, bIf(x, False, True)));
// forall : Pi (A : Type u), (A -> Bool) -> Bool
expr A_pred = A >> Bool;
expr q_type = Pi({A, TypeU}, A_pred >> Bool);
env.add_var(forall_fn_name, q_type);
env.add_definition(exists_fn_name, q_type, Fun({{A,TypeU}, {P, A_pred}}, Not(Forall(A, Fun({x, A}, Not(P(x)))))));
@ -207,6 +211,9 @@ void add_basic_theory(environment & env) {
// 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, Pi({{A, TypeU}, {P, A_pred}, {a, A}, {b, A}, {H1, P(a)}, {H2, Eq(a,b)}}, P(b)));
// eta : Pi (A : Type u) (B : A -> Type u), f : (Pi x : A, B x), (Fun x : A => f x) = f
env.add_axiom(eta_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}}, Eq(Fun({x, A}, f(x)), f)));
// 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_theorem(symm_fn_name, Pi({{A, TypeU}, {a, A}, {b, A}, {H, Eq(a, b)}}, Eq(b, a)),
@ -219,15 +226,12 @@ void add_basic_theory(environment & env) {
Fun({{A, TypeU}, {a, A}, {b, A}, {c, A}, {H1, Eq(a,b)}, {H2, Eq(b,c)}},
Subst(A, Fun({x, A}, Eq(a, x)), b, c, H1, H2)));
// xtrans: Pi (A: Type u) (B : Type u) (a : A) (b c : B) (H1 : a = b) (H2 : b = c), a = c :=
// Subst B (Fun x : B => a = x) b c H1 H2
env.add_theorem(xtrans_fn_name, Pi({{A, TypeU}, {B, TypeU}, {a, A}, {b, B}, {c, B}, {H1, Eq(a, b)}, {H2, Eq(b, c)}}, Eq(a, c)),
Fun({{A, TypeU}, {B, TypeU}, {a, A}, {b, B}, {c, B}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
Subst(B, Fun({x, B}, Eq(a, x)), b, c, H1, H2)));
expr piABx = Pi({x, A}, B(x));
expr A_arrow_u = A >> TypeU;
// congr1 : Pi (A : Type u) (B : A -> Type u) (f g: Pi (x : A) B x) (a : A) (H : f = g), f a = g a :=
// Subst piABx (Fun h : piABx => f a = h a) f g (Refl piABx f) H
env.add_theorem(congr1_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {H, Eq(f, g)}}, Eq(f(a), g(a))),

5
src/kernel/builtin.h
View file

@ -113,6 +113,11 @@ bool is_subst_fn(expr const & e);
/** \brief (Axiom) A : Type u, P : A -> Bool, a b : A, H1 : P a, H2 : a = b |- Subst(A, P, a, b, H1, H2) : P b */
inline expr Subst(expr const & A, expr const & P, expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app({mk_subst_fn(), A, P, a, b, H1, H2}); }
expr mk_eta_fn();
bool is_eta_fn(expr const & e);
/** \brief (Axiom) A : Type u, B : A -> Type u, f : (Pi x : A, B x) |- Eta(A, B, f) : ((Fun x : A => f x) = f) */
inline expr Eta(expr const & A, expr const & B, expr const & f) { return mk_app(mk_eta_fn(), A, B, f); }
expr mk_symm_fn();
bool is_symm_fn(expr const & e);
/** \brief (Theorem) A : Type u, a b : A, H : a = b |- Symm(A, a, b, H) : b = a */