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feat(kernel): heterogeneous transitivity axiom, we need this axiom to be able to generate modular proofs in the rewriting engine module

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-10-29 17:07:30 -07:00
parent a57ca284ec
commit 7fc87faa8f
4 changed files with 22 additions and 19 deletions
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12
src/kernel/builtin.cpp
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@ -160,8 +160,9 @@ MK_CONSTANT(homo_eq_fn, name("eq"));
// Axioms
MK_CONSTANT(mp_fn, name("MP"));
MK_CONSTANT(discharge_fn, name("Discharge"));
MK_CONSTANT(refl_fn, name("Refl"));
MK_CONSTANT(case_fn, name("Case"));
MK_CONSTANT(refl_fn, name("Refl"));
MK_CONSTANT(trans_ext_fn, name("TransExt"));
MK_CONSTANT(subst_fn, name("Subst"));
MK_CONSTANT(eta_fn, name("Eta"));
MK_CONSTANT(imp_antisym_fn, name("ImpAntisym"));
@ -175,11 +176,13 @@ void import_basic(environment & env) {
expr f = Const("f");
expr a = Const("a");
expr b = Const("b");
expr c = Const("c");
expr x = Const("x");
expr y = Const("y");
expr A = Const("A");
expr A_pred = A >> Bool;
expr B = Const("B");
expr C = Const("C");
expr q_type = Pi({A, TypeU}, A_pred >> Bool);
expr piABx = Pi({x, A}, B(x));
expr A_arrow_u = A >> TypeU;
@ -222,11 +225,14 @@ void import_basic(environment & env) {
// Discharge : Pi (a b : Bool) (H : a -> b), a => b
env.add_axiom(discharge_fn_name, Pi({{a, Bool}, {b, Bool}, {H, a >> b}}, Implies(a, b)));
// Case : Pi (P : Bool -> Bool) (H1 : P True) (H2 : P False) (a : Bool), P a
env.add_axiom(case_fn_name, Pi({{P, Bool >> Bool}, {H1, P(True)}, {H2, P(False)}, {a, Bool}}, P(a)));
// Refl : Pi (A : Type u) (a : A), a = a
env.add_axiom(refl_fn_name, Pi({{A, TypeU}, {a, A}}, Eq(a, a)));
// Case : Pi (P : Bool -> Bool) (H1 : P True) (H2 : P False) (a : Bool), P a
env.add_axiom(case_fn_name, Pi({{P, Bool >> Bool}, {H1, P(True)}, {H2, P(False)}, {a, Bool}}, P(a)));
// TransExt : Pi (A B C: Type u) (a : A) (b : B) (c : C) (H1 : a = b) (H2 : b = c), a = c
env.add_axiom(trans_ext_fn_name, Pi({{A, TypeU}, {B, TypeU}, {C, TypeU}, {a, A}, {b, B}, {c, C}, {H1, Eq(a, b)}, {H2, Eq(b, c)}}, Eq(a, c)));
// Subst : Pi (A : Type u) (a b : A) (P : A -> bool) (H1 : P a) (H2 : a = b), P b
env.add_axiom(subst_fn_name, Pi({{A, TypeU}, {a, A}, {b, A}, {P, A_pred}, {H1, P(a)}, {H2, Eq(a, b)}}, P(b)));

17
src/kernel/builtin.h
View file

@ -123,21 +123,28 @@ expr mk_discharge_fn();
/** \brief (Axiom) {a : Bool}, {b : Bool}, H : a -> b |- Discharge(a, b, H) : a => b */
inline expr Discharge(expr const & a, expr const & b, expr const & H) { return mk_app(mk_discharge_fn(), a, b, H); }
/** \brief Reflexivity axiom */
expr mk_refl_fn();
/** \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); }
/** \brief Case analysis axiom */
expr mk_case_fn();
/** \brief (Axiom) P : Bool -> Bool, H1 : P True, H2 : P False, a : Bool |- Case(P, H1, H2, a) : P a */
inline expr Case(expr const & P, expr const & H1, expr const & H2, expr const & a) { return mk_app(mk_case_fn(), P, H1, H2, a); }
/** \brief Reflexivity axiom */
expr mk_refl_fn();
/** \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); }
/** \brief Substitution axiom */
expr mk_subst_fn();
/** \brief (Axiom) {A : Type u}, {a b : A}, P : A -> Bool, H1 : P a, H2 : a = b |- Subst(A, a, b, P, H1, H2) : P b */
inline expr Subst(expr const & A, expr const & a, expr const & b, expr const & P, expr const & H1, expr const & H2) { return mk_app({mk_subst_fn(), A, a, b, P, H1, H2}); }
/** \brief Heterogeneous Transitivity axiom */
expr mk_trans_ext_fn();
/** \brief (Axiom) {A : Type u}, {B : Type u}, {B : Type u}, {a : A}, {b : B} {c : C}, H1 : a = b, H2 : b = c |- TransExt(A, B, a, b, c, H1, H2) : a = c */
inline expr TransExt(expr const & A, expr const & B, expr const & C, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) {
return mk_app({mk_trans_ext_fn(), A, B, C, a, b, c, H1, H2});
}
/** \brief Eta conversion axiom */
expr mk_eta_fn();
/** \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) */

8
src/library/basic_thms.cpp
View file

@ -32,7 +32,6 @@ MK_CONSTANT(disj2_fn, name("Disj2"));
MK_CONSTANT(disj_cases_fn, name("DisjCases"));
MK_CONSTANT(symm_fn, name("Symm"));
MK_CONSTANT(trans_fn, name("Trans"));
MK_CONSTANT(trans_ext_fn, name("TransExt"));
MK_CONSTANT(congr1_fn, name("Congr1"));
MK_CONSTANT(congr2_fn, name("Congr2"));
MK_CONSTANT(congr_fn, name("Congr"));
@ -207,11 +206,6 @@ void import_basic_thms(environment & env) {
Fun({{A, TypeU}, {a, A}, {b, A}, {c, A}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
Subst(A, b, c, Fun({x, A}, Eq(a, x)), H1, H2)));
// TransExt: Pi (A: Type u) (B : Type u) (a : A) (b c : B) (H1 : a = b) (H2 : b = c), a = c
env.add_theorem(trans_ext_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, b, c, Fun({x, B}, Eq(a, x)), H1, H2)));
// EqTElim : Pi (a : Bool) (H : a = True), a
env.add_theorem(eqt_elim_fn_name, Pi({{a, Bool}, {H, Eq(a, True)}}, a),
Fun({{a, Bool}, {H, Eq(a, True)}},
@ -258,7 +252,7 @@ void import_basic_thms(environment & env) {
// 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
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)}},
TransExt(B(a), B(b), f(a), f(b), g(b),
TransExt(B(a), B(b), B(b), f(a), f(b), g(b),
Congr2(A, B, a, b, f, H2), Congr1(A, B, f, g, b, H1))));

4
src/library/basic_thms.h
View file

@ -92,10 +92,6 @@ expr mk_trans_fn();
/** \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_trans_ext_fn();
/** \brief (Theorem) {A : Type u}, {B : Type u}, {a : A}, {b c : B}, H1 : a = b, H2 : b = c |- TransExt(A, B, a, b, c, H1, H2) : a = c */
inline expr TransExt(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_trans_ext_fn(), A, B, a, b, c, H1, H2}); }
expr mk_eqt_elim_fn();
/** \brief (Theorem) {a : Bool}, H : a = True |- EqTElim(a, H) : a */
inline expr EqTElim(expr const & a, expr const & H) { return mk_app(mk_eqt_elim_fn(), a, H); }