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Prove congr1, congr2 and congr theorems. Add xtrans theorem.

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-08-06 14:37:30 -07:00
parent ab915fb3f0
commit 68d092f5ef
2 changed files with 62 additions and 3 deletions
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34
src/kernel/builtin.cpp
View file

@ -151,6 +151,9 @@ 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(xtrans_fn, name("xtrans"));
MK_CONSTANT(congr1_fn, name("congr1"));
MK_CONSTANT(congr2_fn, name("congr2"));
MK_CONSTANT(congr_fn, name("congr"));
MK_CONSTANT(eq_mp_fn, name("eq_mp"));
MK_CONSTANT(truth, name("truth"));
@ -170,13 +173,14 @@ void add_basic_theory(environment & env) {
expr A = Const("A");
expr a = Const("a");
expr b = Const("b");
expr c = Const("a");
expr c = Const("c");
expr H = Const("H");
expr H1 = Const("H1");
expr H2 = Const("H2");
expr B = Const("B");
expr f = Const("f");
expr g = Const("g");
expr h = Const("h");
expr x = Const("x");
expr y = Const("y");
expr P = Const("P");
@ -199,6 +203,7 @@ void add_basic_theory(environment & env) {
// refl : Pi (A : Type u) (a : A), a = a
env.add_axiom(refl_fn_name, Pi({{A, TypeU}, {a, A}}, Eq(a, a)));
// 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)));
@ -214,10 +219,33 @@ 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)));
// 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
// 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;
env.add_axiom(congr_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {b, A}, {H1, Eq(f, g)}, {H2, Eq(a, b)}}, Eq(f(a), g(b))));
// 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))),
Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {H, Eq(f, g)}},
Subst(piABx, Fun({h, piABx}, Eq(f(a), h(a))), f, g, Refl(piABx, f), H)));
// congr2 : Pi (A : Type u) (B : A -> Type u) (f : Pi (x : A) B x) (a b : A) (H : a = b), f a = f b :=
// Subst A (Fun x : A => f a = f x) a b (Refl A a) H
env.add_theorem(congr2_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {a, A}, {b, A}, {H, Eq(a, b)}}, Eq(f(a), f(b))),
Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {a, A}, {b, A}, {H, Eq(a, b)}},
Subst(A, Fun({x, A}, Eq(f(a), f(x))), a, b, Refl(A, a), H)));
// 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 :=
// xTrans (B a) (B b) (f a) (f b) (g b) (congr2 A B f g b H1) (congr1 A B f a b H2)
env.add_theorem(congr_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {b, A}, {H1, Eq(f, g)}, {H2, Eq(a, b)}}, Eq(f(a), g(b))),
Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {b, A}, {H1, Eq(f, g)}, {H2, Eq(a, b)}},
xTrans(B(a), B(b), f(a), f(b), g(b),
Congr2(A, B, f, a, b, H2), Congr1(A, B, f, g, b, H1))));
// eq_mp : Pi (a b: Bool) (H1 : a = b) (H2 : a), b :=
// Subst Bool (Fun x : Bool => x) a b H2 H1

31
src/kernel/builtin.h
View file

@ -105,23 +105,54 @@ inline expr Exists(expr const & A, expr const & P) { return mk_exists(A, P); }
expr mk_refl_fn();
bool is_refl_fn(expr const & e);
/** \brief (Axiom) A : Type u, a : A |- Refl(A, a) : a = a */
inline expr Refl(expr const & A, expr const & a) { return mk_app(mk_refl_fn(), A, a); }
expr mk_subst_fn();
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_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 */
inline expr Symm(expr const & A, expr const & a, expr const & b, expr const & H) { return mk_app(mk_symm_fn(), A, a, b, H); }
expr mk_trans_fn();
bool is_trans_fn(expr const & e);
/** \brief (Theorem) A : Type u, a b c : A, H1 : a = b, H2 : b = c |- Trans(A, a, b, c, H1, H2) : a = c */
inline expr Trans(expr const & A, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) { return mk_app({mk_trans_fn(), A, a, b, c, H1, H2}); }
expr mk_xtrans_fn();
bool is_xtrans_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : Type u, a : A, b c : B, H1 : a = b, H2 : b = c |- xTrans(A, B, a, b, c, H1, H2) : a = c */
inline expr xTrans(expr const & A, expr const & B, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) { return mk_app({mk_xtrans_fn(), A, B, a, b, c, H1, H2}); }
expr mk_congr1_fn();
bool is_congr1_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f g : (Pi x : A, B x), a : A, H : f = g |- Congr2(A, B, f, g, a, H) : f a = g a */
inline expr Congr1(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & H) { return mk_app({mk_congr1_fn(), A, B, f, g, a, H}); }
expr mk_congr2_fn();
bool is_congr2_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f : (Pi x : A, B x), a b : A, H : a = b |- Congr1(A, B, f, a, b, H) : f a = f b */
inline expr Congr2(expr const & A, expr const & B, expr const & f, expr const & a, expr const & b, expr const & H) { return mk_app({mk_congr2_fn(), A, B, f, a, b, H}); }
expr mk_congr_fn();
bool is_congr_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f g : (Pi x : A, B x), a b : A, H1 : f = g, H2 : a = b |- Congr(A, B, f, g, a, b, H1, H2) : f a = g b */
inline expr Congr(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app({mk_congr_fn(), A, B, f, g, a, b, H1, H2}); }
expr mk_eq_mp_fn();
bool is_eq_mp_fn(expr const & e);
/** \brief (Theorem) a : Bool, b : Bool, H1 : a = b, H2 : a |- EqMP(a, b, H1, H2) : b */
inline expr EqMP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_eq_mp_fn(), a, b, H1, H2); }
expr mk_truth();
bool is_truth(expr const & e);
/** \brief (Theorem) Truth : True */
#define Truth mk_truth()
expr mk_ext_fn();
bool is_ext_fn(expr const & e);
expr mk_foralle_fn();