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feat(library/norm_num): extend norm_num to handle subtraction

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Rob Lewis 2015-10-22 18:14:00 -04:00 committed by Leonardo de Moura
parent 8f378db661
commit 616450be64
7 changed files with 462 additions and 53 deletions
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40
library/algebra/numeral.lean
View file

@ -136,3 +136,43 @@ theorem mk_cong (op : A → A) (a b : A) (H : a = b) : op a = op b :=
by congruence; exact H
theorem mk_eq (a : A) : a = a := rfl
theorem neg_add_neg_eq_of_add_add_eq_zero [s : add_comm_group A] (a b c : A) (H : c + a + b = 0) : -a + -b = c :=
begin apply add_neg_eq_of_eq_add, apply neg_eq_of_add_eq_zero, rewrite [add.comm, add.assoc, add.comm b, -add.assoc, H] end
/-theorem neg_add_neg_helper [s : add_comm_group A] (t₁ t₂ e w₁ w₂ : A) (H₁ : t₁ = -w₁)
(H₂ : t₂ = -w₂) (H : e + w₁ + w₂ = 0) : t₁ + t₂ = e :=
by rewrite [H₁, H₂, neg_add_neg_eq_of_add_add_eq_zero _ _ _ H]-/
theorem neg_add_neg_helper [s : add_comm_group A] (a b c : A) (H : a + b = c) : -a + -b = -c :=
begin apply iff.mp !neg_eq_neg_iff_eq, rewrite [neg_add, *neg_neg, H] end
theorem neg_add_pos_eq_of_eq_add [s : add_comm_group A] (a b c : A) (H : b = c + a) : -a + b = c :=
begin apply neg_add_eq_of_eq_add, rewrite add.comm, exact H end
/-theorem neg_add_pos_helper [s : add_comm_group A] (t₁ t₂ e v w₁ w₂ : A) (H₁ : t₁ = -w₁)
(H₂ : t₂ = w₂) (Hv : w₂ = v) (H : e + w₁ = v) : t₁ + t₂ = e :=
begin rewrite [H₁, H₂, Hv, -H, add.comm, add_neg_cancel_right] end-/
theorem neg_add_pos_helper1 [s : add_comm_group A] (a b c : A) (H : b + c = a) : -a + b = -c :=
begin apply neg_add_eq_of_eq_add, apply eq_add_neg_of_add_eq H end
theorem neg_add_pos_helper2 [s : add_comm_group A] (a b c : A) (H : a + c = b) : -a + b = c :=
begin apply neg_add_eq_of_eq_add, rewrite H end
theorem pos_add_neg_helper [s : add_comm_group A] (a b c : A) (H : b + a = c) : a + b = c :=
by rewrite [add.comm, H]
theorem sub_eq_add_neg_helper [s : add_comm_group A] (t₁ t₂ e w₁ w₂: A) (H₁ : t₁ = w₁) (H₂ : t₂ = w₂)
(H : w₁ + -w₂ = e) : t₁ - t₂ = e :=
by rewrite [sub_eq_add_neg, H₁, H₂, H]
theorem pos_add_pos_helper [s : add_comm_group A] (a b c h₁ h₂ : A) (H₁ : a = h₁) (H₂ : b = h₂)
(H : h₁ + h₂ = c) : a + b = c :=
by rewrite [H₁, H₂, H]
theorem subst_into_subtr [s : add_group A] (l r t : A) (prt : l + -r = t) : l - r = t :=
by rewrite [sub_eq_add_neg, prt]
theorem neg_neg_helper [s : add_group A] (a b : A) (H : a = -b) : -a = b :=
by rewrite [H, neg_neg]

430
src/library/norm_num.cpp
View file

@ -6,11 +6,13 @@ Author: Robert Y. Lewis
#include "library/norm_num.h"
#include "library/constants.h"
namespace lean {
static name * g_add = nullptr,
* g_add1 = nullptr,
* g_mul = nullptr,
* g_sub = nullptr,
* g_neg = nullptr,
* g_bit0_add_bit0 = nullptr,
* g_bit1_add_bit0 = nullptr,
* g_bit0_add_bit1 = nullptr,
@ -28,6 +30,7 @@ static name * g_add = nullptr,
* g_add1_zero = nullptr,
* g_add1_one = nullptr,
* g_subst_sum = nullptr,
* g_subst_subtr = nullptr,
* g_subst_prod = nullptr,
* g_mk_cong = nullptr,
* g_mk_eq = nullptr,
@ -40,9 +43,21 @@ static name * g_add = nullptr,
* g_add_monoid = nullptr,
* g_monoid = nullptr,
* g_add_comm = nullptr,
* g_add_group = nullptr,
* g_mul_zero_class = nullptr,
* g_distrib = nullptr,
* g_semiring = nullptr;
* g_has_neg = nullptr,
* g_has_sub = nullptr,
* g_semiring = nullptr,
* g_eq_neg_of_add_eq_zero = nullptr,
* g_neg_add_neg_eq = nullptr,
* g_neg_add_pos1 = nullptr,
* g_neg_add_pos2 = nullptr,
* g_pos_add_neg = nullptr,
* g_pos_add_pos = nullptr,
* g_sub_eq_add_neg = nullptr,
* g_neg_neg = nullptr,
* g_add_comm_group = nullptr;
static bool is_numeral_aux(expr const & e, bool is_first) {
buffer<expr> args;
@ -64,13 +79,15 @@ bool norm_num_context::is_numeral(expr const & e) const {
return is_numeral_aux(e, true);
}
bool is_neg(expr const & e) {
return is_const_app(e, *g_neg, 3);
}
/*
Takes e : instance A, and tries to synthesize has_add A.
Takes A : Type, and tries to synthesize has_add A.
*/
expr norm_num_context::mk_has_add(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(get_has_add_name(), const_levels(f)), args[0]);
expr t = mk_app(mk_constant(get_has_add_name(), m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -80,9 +97,7 @@ expr norm_num_context::mk_has_add(expr const & e) {
}
expr norm_num_context::mk_has_mul(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_has_mul, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_has_mul, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -92,9 +107,17 @@ expr norm_num_context::mk_has_mul(expr const & e) {
}
expr norm_num_context::mk_has_one(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(get_has_one_name(), const_levels(f)), args[0]);
expr t = mk_app(mk_constant(get_has_one_name(), m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
} else {
throw exception("failed to synthesize has_one instance");
}
}
expr norm_num_context::mk_has_zero(expr const & e) {
expr t = mk_app(mk_constant(get_has_zero_name(), m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -104,9 +127,7 @@ expr norm_num_context::mk_has_one(expr const & e) {
}
expr norm_num_context::mk_add_monoid(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_add_monoid, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_add_monoid, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -116,9 +137,7 @@ expr norm_num_context::mk_add_monoid(expr const & e) {
}
expr norm_num_context::mk_monoid(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_monoid, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_monoid, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -128,9 +147,17 @@ expr norm_num_context::mk_monoid(expr const & e) {
}
expr norm_num_context::mk_add_comm(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_add_comm, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_add_comm, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
} else {
throw exception("failed to synthesize add_comm_semigroup instance");
}
}
expr norm_num_context::mk_add_group(expr const & e) {
expr t = mk_app(mk_constant(*g_add_group, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -140,9 +167,7 @@ expr norm_num_context::mk_add_comm(expr const & e) {
}
expr norm_num_context::mk_has_distrib(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_distrib, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_distrib, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -152,9 +177,7 @@ expr norm_num_context::mk_has_distrib(expr const & e) {
}
expr norm_num_context::mk_mul_zero_class(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_mul_zero_class, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_mul_zero_class, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -164,9 +187,7 @@ expr norm_num_context::mk_mul_zero_class(expr const & e) {
}
expr norm_num_context::mk_semiring(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
expr t = mk_app(mk_constant(*g_semiring, const_levels(f)), args[0]);
expr t = mk_app(mk_constant(*g_semiring, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
@ -175,6 +196,36 @@ expr norm_num_context::mk_semiring(expr const & e) {
}
}
expr norm_num_context::mk_has_neg(expr const & e) {
expr t = mk_app(mk_constant(*g_has_neg, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
} else {
throw exception("failed to synthesize has_neg instance");
}
}
expr norm_num_context::mk_has_sub(expr const & e) {
expr t = mk_app(mk_constant(*g_has_sub, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
} else {
throw exception("failed to synthesize has_sub instance");
}
}
expr norm_num_context::mk_add_comm_group(expr const & e) {
expr t = mk_app(mk_constant(*g_add_comm_group, m_lvls), e);
optional<expr> inst = mk_class_instance(m_env, m_ctx, t);
if (inst) {
return *inst;
} else {
throw exception("failed to synthesize add_comm_group instance");
}
}
expr norm_num_context::mk_const(name const & n) {
return mk_constant(n, m_lvls);
}
@ -183,7 +234,7 @@ expr norm_num_context::mk_cong(expr const & op, expr const & type, expr const &
return mk_app({mk_const(*g_mk_cong), type, op, a, b, eq});
}
pair<expr, expr> norm_num_context::mk_norm(expr const & e) {
/*pair<expr, expr> norm_num_context::mk_norm(expr const & e) {
buffer<expr> args;
expr f = get_app_args(e, args);
if (!is_constant(f)) {
@ -194,14 +245,14 @@ pair<expr, expr> norm_num_context::mk_norm(expr const & e) {
auto lhs_p = mk_norm(args[2]);
auto rhs_p = mk_norm(args[3]);
auto add_p = mk_norm_add(lhs_p.first, rhs_p.first);
expr prf = mk_app({mk_const(*g_subst_sum), args[0], mk_has_add(args[1]), args[2], args[3],
expr prf = mk_app({mk_const(*g_subst_sum), args[0], mk_has_add(args[0]), args[2], args[3],
lhs_p.first, rhs_p.first, add_p.first, lhs_p.second, rhs_p.second, add_p.second});
return pair<expr, expr>(add_p.first, prf);
} else if (const_name(f) == *g_mul && args.size() == 4) {
auto lhs_p = mk_norm(args[2]);
auto rhs_p = mk_norm(args[3]);
auto mul_p = mk_norm_mul(lhs_p.first, rhs_p.first);
expr prf = mk_app({mk_const(*g_subst_prod), args[0], mk_has_mul(args[1]), args[2], args[3],
expr prf = mk_app({mk_const(*g_subst_prod), args[0], mk_has_mul(args[0]), args[2], args[3],
lhs_p.first, rhs_p.first, mul_p.first, lhs_p.second, rhs_p.second, mul_p.second});
return pair<expr, expr>(mul_p.first, prf);
} else if (const_name(f) == get_bit0_name() && args.size() == 3) {
@ -221,7 +272,7 @@ pair<expr, expr> norm_num_context::mk_norm(expr const & e) {
throw exception("mk_norm found unrecognized combo ");
}
// TODO(Rob): cases for sub, div
}
}*/
// returns <t, p> such that p is a proof that lhs + rhs = t.
pair<expr, expr> norm_num_context::mk_norm_add(expr const & lhs, expr const & rhs) {
@ -234,29 +285,30 @@ pair<expr, expr> norm_num_context::mk_norm_add(expr const & lhs, expr const & rh
}
auto type = args_lhs[0];
auto typec = args_lhs[1];
// std::cout << "typec in mk_norm_add is: " << typec << ". lhs: " << lhs << ", rhs: " << rhs << "\n";
expr rv;
expr prf;
if (is_bit0(lhs) && is_bit0(rhs)) { // typec is has_add
auto p = mk_norm_add(args_lhs[2], args_rhs[2]);
rv = mk_app(lhs_head, type, typec, p.first);
prf = mk_app({mk_const(*g_bit0_add_bit0), type, mk_add_comm(typec), args_lhs[2], args_rhs[2], p.first, p.second});
prf = mk_app({mk_const(*g_bit0_add_bit0), type, mk_add_comm(type), args_lhs[2], args_rhs[2], p.first, p.second});
} else if (is_bit0(lhs) && is_bit1(rhs)) {
auto p = mk_norm_add(args_lhs[2], args_rhs[3]);
rv = mk_app({rhs_head, type, args_rhs[1], args_rhs[2], p.first});
prf = mk_app({mk_const(*g_bit0_add_bit1), type, mk_add_comm(typec), args_rhs[1], args_lhs[2], args_rhs[3], p.first, p.second});
prf = mk_app({mk_const(*g_bit0_add_bit1), type, mk_add_comm(type), args_rhs[1], args_lhs[2], args_rhs[3], p.first, p.second});
} else if (is_bit0(lhs) && is_one(rhs)) {
rv = mk_app({mk_const(get_bit1_name()), type, args_rhs[1], args_lhs[1], args_lhs[2]});
prf = mk_app({mk_const(*g_bit0_add_1), type, typec, args_rhs[1], args_lhs[2]});
} else if (is_bit1(lhs) && is_bit0(rhs)) { // typec is has_one
auto p = mk_norm_add(args_lhs[3], args_rhs[2]);
rv = mk_app(lhs_head, type, typec, args_lhs[2], p.first);
prf = mk_app({mk_const(*g_bit1_add_bit0), type, mk_add_comm(typec), typec, args_lhs[3], args_rhs[2], p.first, p.second});
prf = mk_app({mk_const(*g_bit1_add_bit0), type, mk_add_comm(type), typec, args_lhs[3], args_rhs[2], p.first, p.second});
} else if (is_bit1(lhs) && is_bit1(rhs)) { // typec is has_one
auto add_ts = mk_norm_add(args_lhs[3], args_rhs[3]);
expr add1 = mk_app({mk_const(*g_add1), type, args_lhs[2], typec, add_ts.first});
auto p = mk_norm_add1(add1);
rv = mk_app({mk_const(get_bit0_name()), type, args_lhs[2], p.first});
prf = mk_app({mk_const(*g_bit1_add_bit1), type, mk_add_comm(typec), typec, args_lhs[3], args_rhs[3], add_ts.first, p.first, add_ts.second, p.second});
prf = mk_app({mk_const(*g_bit1_add_bit1), type, mk_add_comm(type), typec, args_lhs[3], args_rhs[3], add_ts.first, p.first, add_ts.second, p.second});
} else if (is_bit1(lhs) && is_one(rhs)) { // typec is has_one
expr add1 = mk_app({mk_const(*g_add1), type, args_lhs[2], typec, lhs});
auto p = mk_norm_add1(add1);
@ -264,21 +316,21 @@ pair<expr, expr> norm_num_context::mk_norm_add(expr const & lhs, expr const & rh
prf = mk_app({mk_const(*g_bit1_add_1), type, args_lhs[2], typec, args_lhs[3], p.first, p.second});
} else if (is_one(lhs) && is_bit0(rhs)) { // typec is has_one
rv = mk_app({mk_const(get_bit1_name()), type, typec, args_rhs[1], args_rhs[2]});
prf = mk_app({mk_const(*g_1_add_bit0), type, mk_add_comm(typec), typec, args_rhs[2]});
prf = mk_app({mk_const(*g_1_add_bit0), type, mk_add_comm(type), typec, args_rhs[2]});
} else if (is_one(lhs) && is_bit1(rhs)) { // typec is has_one
expr add1 = mk_app({mk_const(*g_add1), type, args_rhs[2], args_rhs[1], rhs});
auto p = mk_norm_add1(add1);
rv = p.first;
prf = mk_app({mk_const(*g_1_add_bit1), type, mk_add_comm(typec), typec, args_rhs[3], p.first, p.second});
prf = mk_app({mk_const(*g_1_add_bit1), type, mk_add_comm(type), typec, args_rhs[3], p.first, p.second});
} else if (is_one(lhs) && is_one(rhs)) {
rv = mk_app({mk_const(get_bit0_name()), type, mk_has_add(typec), lhs});
prf = mk_app({mk_const(*g_one_add_one), type, mk_has_add(typec), typec});
rv = mk_app({mk_const(get_bit0_name()), type, mk_has_add(type), lhs});
prf = mk_app({mk_const(*g_one_add_one), type, mk_has_add(type), typec});
} else if (is_zero(lhs)) {
rv = rhs;
prf = mk_app({mk_const(*g_bin_0_add), type, mk_add_monoid(typec), rhs});
prf = mk_app({mk_const(*g_bin_0_add), type, mk_add_monoid(type), rhs});
} else if (is_zero(rhs)) {
rv = lhs;
prf = mk_app({mk_const(*g_bin_add_0), type, mk_add_monoid(typec), lhs});
prf = mk_app({mk_const(*g_bin_add_0), type, mk_add_monoid(type), lhs});
} else {
std::cout << "\n\n bad args: " << lhs_head << ", " << rhs_head << "\n";
throw exception("mk_norm_add got malformed args");
@ -302,10 +354,10 @@ pair<expr, expr> norm_num_context::mk_norm_add1(expr const & e) {
} else if (is_bit1(p)) { // ne_args : has_one, has_add
auto np = mk_norm_add1(mk_app({mk_const(*g_add1), args[0], args[1], args[2], ne_args[3]}));
rv = mk_app({mk_const(get_bit0_name()), args[0], args[1], np.first});
prf = mk_app({mk_const(*g_add1_bit1), args[0], mk_add_comm(args[1]), args[2], ne_args[3], np.first, np.second});
prf = mk_app({mk_const(*g_add1_bit1), args[0], mk_add_comm(args[0]), args[2], ne_args[3], np.first, np.second});
} else if (is_zero(p)) {
rv = mk_app({mk_const(get_one_name()), args[0], args[2]});
prf = mk_app({mk_const(*g_add1_zero), args[0], mk_add_monoid(args[1]), args[2]});
prf = mk_app({mk_const(*g_add1_zero), args[0], mk_add_monoid(args[0]), args[2]});
} else if (is_one(p)) {
rv = mk_app({mk_const(get_bit0_name()), args[0], args[1], mk_app({mk_const(get_one_name()), args[0], args[2]})});
prf = mk_app({mk_const(*g_add1_one), args[0], args[1], args[2]});
@ -330,22 +382,22 @@ pair<expr, expr> norm_num_context::mk_norm_mul(expr const & lhs, expr const & rh
expr prf;
if (is_zero(rhs)) {
rv = rhs;
prf = mk_app({mk_const(*g_mul_zero), type, mk_mul_zero_class(typec), lhs});
prf = mk_app({mk_const(*g_mul_zero), type, mk_mul_zero_class(type), lhs});
} else if (is_zero(lhs)) {
rv = lhs;
prf = mk_app({mk_const(*g_zero_mul), type, mk_mul_zero_class(typec), rhs});
prf = mk_app({mk_const(*g_zero_mul), type, mk_mul_zero_class(type), rhs});
} else if (is_one(rhs)) {
rv = lhs;
prf = mk_app({mk_const(*g_mul_one), type, mk_monoid(typec), lhs});
prf = mk_app({mk_const(*g_mul_one), type, mk_monoid(type), lhs});
} else if (is_bit0(rhs)) {
auto mtp = mk_norm_mul(lhs, args_rhs[2]);
rv = mk_app({rhs_head, type, typec, mtp.first});
prf = mk_app({mk_const(*g_mul_bit0), type, mk_has_distrib(typec), lhs, args_rhs[2], mtp.first, mtp.second});
prf = mk_app({mk_const(*g_mul_bit0), type, mk_has_distrib(type), lhs, args_rhs[2], mtp.first, mtp.second});
} else if (is_bit1(rhs)) {
auto mtp = mk_norm_mul(lhs, args_rhs[3]);
auto atp = mk_norm_add(mk_app({mk_const(get_bit0_name()), type, args_rhs[2], mtp.first}), lhs);
rv = atp.first;
prf = mk_app({mk_const(*g_mul_bit1), type, mk_semiring(typec), lhs, args_rhs[3],
prf = mk_app({mk_const(*g_mul_bit1), type, mk_semiring(type), lhs, args_rhs[3],
mtp.first, atp.first, mtp.second, atp.second});
} else {
std::cout << "bad args to mk_norm_mul: " << rhs << "\n";
@ -364,11 +416,262 @@ pair<expr, expr> norm_num_context::mk_norm_sub(expr const &, expr const &) {
throw exception("not implemented yet -- mk_norm_sub");
}
mpz norm_num_context::num_of_expr(expr const & e) { // note : mpz only supports nonneg nums
buffer<expr> args;
expr f = get_app_args(e, args);
if (!is_constant(f)) {
throw exception("cannot find num of nonconstant");
}
auto v = to_num(e);
if (v) {
return *v;
}
if (const_name(f) == *g_add && args.size() == 4) {
return num_of_expr(args[2]) + num_of_expr(args[3]);
} else if (const_name(f) == *g_mul && args.size() == 4) {
return num_of_expr(args[2]) * num_of_expr(args[3]);
} else if (const_name(f) == *g_sub && args.size() == 4) {
return num_of_expr(args[2]) - num_of_expr(args[3]);
} else if (const_name(f) == *g_neg && args.size() == 3) {
return neg(num_of_expr(args[2]));
} else {
std::cout << "error : " << f << "\n";
throw exception("expression in num_of_expr is malfomed");
}
}
pair<expr, expr> get_type_and_arg_of_neg(expr & e) {
lean_assert(is_neg(e));
buffer<expr> args;
expr f = get_app_args(e, args);
return pair<expr, expr>(args[0], args[2]);
}
// returns a proof that s_lhs + s_rhs = rhs, where all are negated numerals
expr norm_num_context::mk_norm_eq_neg_add_neg(expr & s_lhs, expr & s_rhs, expr & rhs) {
lean_assert(is_neg(s_lhs));
lean_assert(is_neg(s_rhs));
lean_assert(is_neg(rhs));
auto s_lhs_v = get_type_and_arg_of_neg(s_lhs).second;
auto s_rhs_v = get_type_and_arg_of_neg(s_rhs).second;
auto rhs_v = get_type_and_arg_of_neg(rhs);
expr type = rhs_v.first;
auto sum_pr = mk_norm_eq_pos_add_pos(s_lhs_v, s_rhs_v, rhs_v.second);
return mk_app({mk_const(*g_neg_add_neg_eq), type, mk_add_comm_group(type), s_lhs_v, s_rhs_v, rhs_v.second, sum_pr});
}
expr norm_num_context::mk_norm_eq_neg_add_pos(expr & s_lhs, expr & s_rhs, expr & rhs) {
lean_assert(is_neg(s_lhs));
lean_assert(!is_neg(s_rhs));
auto s_lhs_v = get_type_and_arg_of_neg(s_lhs);
expr type = s_lhs_v.first;
if (is_neg(rhs)) {
auto rhs_v = get_type_and_arg_of_neg(rhs).second;
auto sum_pr = mk_norm_eq_pos_add_pos(s_rhs, rhs_v, s_lhs_v.second);
return mk_app({mk_const(*g_neg_add_pos1), type, mk_add_comm_group(type), s_lhs_v.second, s_rhs, rhs_v, sum_pr});
} else {
auto sum_pr = mk_norm_eq_pos_add_pos(s_lhs_v.second, rhs, s_rhs);
return mk_app({mk_const(*g_neg_add_pos2), type, mk_add_comm_group(type), s_lhs_v.second, s_rhs, rhs, sum_pr});
}
}
expr norm_num_context::mk_norm_eq_pos_add_neg(expr & s_lhs, expr & s_rhs, expr & rhs) {
expr prf = mk_norm_eq_neg_add_pos(s_rhs, s_lhs, rhs);
expr type = get_type_and_arg_of_neg(s_rhs).first;
return mk_app({mk_const(*g_pos_add_neg), type, mk_add_comm_group(type), s_lhs, s_rhs, rhs, prf});
}
// returns a proof that s_lhs + s_rhs = rhs, where all are nonneg normalized numerals
expr norm_num_context::mk_norm_eq_pos_add_pos(expr & s_lhs, expr & s_rhs, expr & rhs) {
lean_assert(!is_neg(s_lhs));
lean_assert(!is_neg(s_rhs));
lean_assert(!is_neg(rhs));
auto p = mk_norm_add(s_lhs, s_rhs);
lean_assert(to_num(rhs) == to_num(p.first));
return p.second;
}
/*expr norm_num_context::mk_norm_eq(expr const & lhs, expr const & rhs) { // rhs is a nonneg numeral
buffer<expr> args;
expr f = get_app_args(lhs, args);
if (!is_constant(f)) {
throw exception("cannot take norm of nonconstant");
}
// m_lvls = const_levels(f);
// expr rv;
// expr prf;
if (const_name(f) == *g_add && args.size() == 4) {
auto lhs_p = num_of_expr(args[2]); // mk_norm_expr(args[2]);
auto rhs_p = num_of_expr(args[3]); //mk_norm_expr(args[3]);
buffer<expr> args_lhs, args_rhs;
// expr flhs = get_app_args(lhs_p.first, args_lhs);
// expr frhs = get_app_args(rhs_p.first, args_rhs);
// std::cout << "in mk_norm_eq add. is_neg first, second:" << is_neg(lhs_p.first) << is_neg(rhs_p.first) << "\n";
if (lhs_p.is_neg()) {
if (rhs_p.is_neg()) {
return mk_norm_eq_neg_add_neg(f, rhs, args);
//return mk_norm_eq_neg_add_neg(f, rhs, args, args_lhs, args_rhs, lhs_p.second, rhs_p.second);
} else {
return mk_norm_eq_neg_add_pos(f, rhs, args);
}
} else {
if (rhs_p.is_neg()) {
buffer<expr> rvargs = buffer<expr>(args);
rvargs[3] = args[2];
rvargs[2] = args[3];
expr commprf = mk_norm_eq_neg_add_pos(f, rhs, rvargs);
// commprf : args[3] + args[2] = rhs
return mk_app({mk_const(*g_pos_add_neg), args[0], mk_add_comm_group(args[0]), args[2], args[3], rhs, commprf});
} else { // nonneg add nonneg
return mk_norm_eq_pos_add_pos(f, rhs, args);
}
}
} else if (const_name(f) == *g_mul && args.size() == 4) {
auto lhs_p = mk_norm_expr(args[2]);
auto rhs_p = mk_norm_expr(args[3]);// TODO(Rob): handle case where either is neg
auto mul_p = mk_norm_mul(lhs_p.first, rhs_p.first);
rv = mul_p.first;
prf = mk_app({mk_const(*g_subst_prod), args[0], mk_has_mul(args[0]), args[2], args[3],
lhs_p.first, rhs_p.first, mul_p.first, lhs_p.second, rhs_p.second, mul_p.second});
} else if (const_name(f) == *g_sub && args.size() == 4) {
// auto lhs_p = mk_norm_expr(args[2]);
// auto rhs_p = mk_norm_expr(args[3]);
expr addneg = mk_app({mk_const(*g_add), args[0], mk_has_add(args[0]), args[2], mk_neg(args[0], args[3])});
expr prf = mk_norm_eq(addneg, rhs); // a + -b = c
return mk_app({mk_const(*g_sub_eq_add_neg), args[0], mk_add_comm_group(args[0]), args[2], args[3], rhs, prf});
} else if ((const_name(f) == get_bit0_name() || const_name(f) == *g_neg) && args.size() == 3) {
auto arg = mk_norm(args[2]);
// rv = mk_app({f, args[0], args[1], arg.first});
return mk_cong(mk_app({f, args[0], args[1]}), args[0], args[2], arg.first, arg.second);
} else if (const_name(f) == get_bit1_name() && args.size() == 4) {
auto arg = mk_norm(args[3]);
// rv = mk_app({f, args[0], args[1], args[2], arg.first});
return mk_cong(mk_app({f, args[0], args[1], args[2]}), args[0], args[3], arg.first, arg.second);
} else if ((const_name(f) == get_zero_name() || const_name(f) == get_one_name()) && args.size() == 2) {
//rv = lhs;
return mk_app({mk_const(*g_mk_eq), args[0], lhs});
} else {
std::cout << "error with name " << const_name(f) << " and size " << args.size() << ".\n";
throw exception("mk_norm found unrecognized combo ");
}
// throw exception("not implemented yet");
}*/
expr norm_num_context::from_pos_num(mpz const & n, expr const & type) {
lean_assert(n > 0);
if (n == 1)
return mk_app({mk_const(get_one_name()), type, mk_has_one(type)});
if (n % mpz(2) == 1)
return mk_app({mk_const(get_bit1_name()), type, mk_has_one(type), mk_has_add(type), from_pos_num(n/2, type)});
else
return mk_app({mk_const(get_bit0_name()), type, mk_has_add(type), from_pos_num(n/2, type)});
}
expr norm_num_context::from_num(mpz const & n, expr const & type) {
expr r;
lean_assert(n >= 0);
if (n == 0)
r = mk_app(mk_const(get_zero_name()), type, mk_has_zero(type));
else
r = from_pos_num(n, type);
lean_assert(*to_num(r) == n);
return r;
}
expr norm_num_context::mk_neg(expr const & type, expr const & e) {
auto has_neg = mk_has_neg(type);
return mk_app({mk_const(*g_neg), type, has_neg, e});
}
expr norm_num_context::mk_add(expr const & type, expr const & e1, expr const & e2) {
auto has_add = mk_has_add(type);
return mk_app({mk_const(*g_add), type, has_add, e1, e2});
}
pair<expr, expr> norm_num_context::mk_norm(expr const & e) {
std::cout << "mk_norm\n";
buffer<expr> args;
expr f = get_app_args(e, args);
if (!is_constant(f) || args.size() == 0) {
throw exception("malformed argument to mk_norm_expr");
}
m_lvls = const_levels(f);
expr type = args[0];
auto val = num_of_expr(e);
expr nval; // e = nval
if (val >= 0) {
nval = from_num(val, type);
} else {
nval = mk_neg(type, from_num(neg(val), type));
}
if (const_name(f) == *g_add && args.size() == 4) {
auto lhs_p = mk_norm(args[2]);
auto rhs_p = mk_norm(args[3]);
expr prf;
if (is_neg(lhs_p.first)) {
if (is_neg(rhs_p.first)) {
prf = mk_norm_eq_neg_add_neg(lhs_p.first, rhs_p.first, nval);
} else {
prf = mk_norm_eq_neg_add_pos(lhs_p.first, rhs_p.first, nval);
}
} else {
if (is_neg(rhs_p.first)) {
prf = mk_norm_eq_pos_add_neg(lhs_p.first, rhs_p.first, nval);
} else {
prf = mk_norm_eq_pos_add_pos(lhs_p.first, rhs_p.first, nval);
}
}
expr rprf = mk_app({mk_const(*g_subst_sum), type, mk_has_add(type), args[2], args[3],
lhs_p.first, rhs_p.first, nval, lhs_p.second, rhs_p.second, prf});
return pair<expr, expr>(nval, rprf);
} else if (const_name(f) == *g_sub && args.size() == 4) {
expr sum = mk_add(args[0], args[2], mk_neg(args[0], args[3]));
auto anprf = mk_norm(sum);
expr rprf = mk_app({mk_const(*g_subst_subtr), type, mk_add_group(type), args[2], args[3], anprf.first, anprf.second});
return pair<expr, expr>(nval, rprf);
} else if (const_name(f) == *g_neg && args.size() == 3) {
auto prf = mk_norm(args[2]);
lean_assert(num_of_expr(prf.first) == neg(val));
if (is_neg(nval)) {
buffer<expr> nval_args;
get_app_args(nval, nval_args);
auto rprf = mk_cong(mk_app(f, args[0], args[1]), type, args[2], nval_args[2], prf.second);
return pair<expr, expr>(nval, rprf);
} else {
auto rprf = mk_app({mk_const(*g_neg_neg), type, mk_add_group(type), args[2], nval, prf.second});
return pair<expr, expr>(nval, rprf);
}
} else if (const_name(f) == get_bit0_name() && args.size() == 3) {
lean_assert(is_bit0(nval));
buffer<expr> nval_args;
get_app_args(nval, nval_args);
auto prf = mk_norm(args[2]);
auto rprf = mk_cong(mk_app(f, args[0], args[1]), type, args[2], nval_args[2], prf.second);
return pair<expr, expr>(nval, rprf);
} else if (const_name(f) == get_bit1_name() && args.size() == 4) {
lean_assert(is_bit1(nval));
buffer<expr> nval_args;
get_app_args(nval, nval_args);
auto prf = mk_norm(args[3]);
auto rprf = mk_cong(mk_app(f, args[0], args[1], args[2]), type, args[3], nval_args[3], prf.second);
return pair<expr, expr>(nval, rprf);
} else if ((const_name(f) == get_zero_name() || const_name(f) == get_one_name()) && args.size() == 2) {
return pair<expr, expr>(e, mk_app({mk_const(*g_mk_eq), args[0], e}));
} else {
std::cout << "error with name " << const_name(f) << " and size " << args.size() << ".\n";
throw exception("mk_norm found unrecognized combo ");
}
}
void initialize_norm_num() {
g_add = new name("add");
g_add1 = new name("add1");
g_mul = new name("mul");
g_sub = new name("sub");
g_neg = new name("neg");
g_bit0_add_bit0 = new name("bit0_add_bit0_helper");
g_bit1_add_bit0 = new name("bit1_add_bit0_helper");
g_bit0_add_bit1 = new name("bit0_add_bit1_helper");
@ -386,6 +689,7 @@ void initialize_norm_num() {
g_add1_zero = new name("add1_zero");
g_add1_one = new name("add1_one");
g_subst_sum = new name("subst_into_sum");
g_subst_subtr = new name("subst_into_subtr");
g_subst_prod = new name("subst_into_prod");
g_mk_cong = new name("mk_cong");
g_mk_eq = new name("mk_eq");
@ -398,9 +702,21 @@ void initialize_norm_num() {
g_add_monoid = new name("algebra", "add_monoid");
g_monoid = new name("algebra", "monoid");
g_add_comm = new name("algebra", "add_comm_semigroup");
g_add_group = new name("algebra", "add_group");
g_mul_zero_class = new name("algebra", "mul_zero_class");
g_distrib = new name("algebra", "distrib");
g_has_neg = new name("has_neg"); //"algebra",
g_has_sub = new name("algebra", "has_sub");
g_semiring = new name("algebra", "semiring");
g_eq_neg_of_add_eq_zero = new name("algebra", "eq_neg_of_add_eq_zero");
g_neg_add_neg_eq = new name("neg_add_neg_helper");
g_neg_add_pos1 = new name("neg_add_pos_helper1");
g_neg_add_pos2 = new name("neg_add_pos_helper2");
g_pos_add_neg = new name("pos_add_neg_helper");
g_sub_eq_add_neg = new name("sub_eq_add_neg_helper");
g_pos_add_pos = new name("pos_add_pos_helper");
g_neg_neg = new name("neg_neg_helper");
g_add_comm_group = new name("algebra", "add_comm_group");
}
void finalize_norm_num() {
@ -408,6 +724,7 @@ void finalize_norm_num() {
delete g_add1;
delete g_mul;
delete g_sub;
delete g_neg;
delete g_bit0_add_bit0;
delete g_bit1_add_bit0;
delete g_bit0_add_bit1;
@ -425,6 +742,7 @@ void finalize_norm_num() {
delete g_add1_zero;
delete g_add1_one;
delete g_subst_sum;
delete g_subst_subtr;
delete g_subst_prod;
delete g_mk_cong;
delete g_mk_eq;
@ -437,8 +755,20 @@ void finalize_norm_num() {
delete g_add_monoid;
delete g_monoid;
delete g_add_comm;
delete g_add_group;
delete g_mul_zero_class;
delete g_distrib;
delete g_has_neg;
delete g_has_sub;
delete g_semiring;
delete g_eq_neg_of_add_eq_zero;
delete g_neg_add_neg_eq;
delete g_neg_add_pos1;
delete g_neg_add_pos2;
delete g_pos_add_neg;
delete g_pos_add_pos;
delete g_sub_eq_add_neg;
delete g_neg_neg;
delete g_add_comm_group;
}
}

23
src/library/norm_num.h
View file

@ -23,21 +23,39 @@ class norm_num_context {
expr mk_cong(expr const &, expr const &, expr const &, expr const &, expr const &);
expr mk_has_add(expr const &);
expr mk_has_mul(expr const &);
expr mk_has_zero(expr const &);
expr mk_has_one(expr const &);
expr mk_add_monoid(expr const &);
expr mk_monoid(expr const &);
expr mk_has_distrib(expr const &);
expr mk_add_comm(expr const &);
expr mk_add_group(expr const &);
expr mk_mul_zero_class(expr const &);
expr mk_semiring(expr const &);
expr mk_has_neg(expr const &);
expr mk_has_sub(expr const &);
expr mk_add(expr const &, expr const &, expr const &);
expr mk_neg(expr const &, expr const &);
expr mk_add_comm_group(expr const &);
expr mk_norm_eq_neg_add_neg(expr &,expr &,expr &);
expr mk_norm_eq_neg_add_pos(expr &, expr &, expr &);
expr mk_norm_eq_pos_add_neg(expr &, expr &, expr &);
expr mk_norm_eq_pos_add_pos(expr &, expr &, expr &);
public:
norm_num_context(environment const & env, local_context const & ctx):m_env(env), m_ctx(ctx) {}
bool is_numeral(expr const & e) const;
pair<expr, expr> mk_norm(expr const & e);
//pair<expr, expr> mk_norm_expr(expr const & e);
expr mk_norm_eq(expr const &, expr const &);
mpz num_of_expr(expr const & e);
expr from_pos_num(mpz const &, expr const &);
expr from_num(mpz const &, expr const &);
};
inline bool is_neg(expr const & e);
inline bool is_numeral(environment const & env, expr const & e) {
return norm_num_context(env, local_context()).is_numeral(e);
}
@ -45,6 +63,11 @@ inline bool is_numeral(environment const & env, expr const & e) {
inline pair<expr, expr> mk_norm_num(environment const & env, local_context const & ctx, expr const & e) {
return norm_num_context(env, ctx).mk_norm(e);
}
inline mpz num_of_expr(environment const & env, local_context const & ctx, expr const & e) {
return norm_num_context(env, ctx).num_of_expr(e);
}
void initialize_norm_num();
void finalize_norm_num();
}

11
src/library/num.cpp
View file

@ -18,7 +18,7 @@ bool has_num_decls(environment const & env) {
env.find(get_bit1_name());
}
static bool is_const_app(expr const & e, name const & n, unsigned nargs) {
bool is_const_app(expr const & e, name const & n, unsigned nargs) {
expr const & f = get_app_fn(e);
return is_constant(f) && const_name(f) == n && get_app_num_args(e) == nargs;
}
@ -43,6 +43,12 @@ optional<expr> is_bit1(expr const & e) {
return some_expr(app_arg(e));
}
optional<expr> is_neg(expr const & e) {
if (!is_const_app(e, *new name("neg"), 3))
return none_expr();
return some_expr(app_arg(e));
}
optional<expr> unfold_num_app(environment const & env, expr const & e) {
if (is_zero(e) || is_one(e) || is_bit0(e) || is_bit1(e)) {
return unfold_app(env, e);
@ -83,6 +89,9 @@ static optional<mpz> to_num(expr const & e, bool first) {
} else if (auto a = is_bit1(e)) {
if (auto r = to_num(*a, false))
return some(2*(*r)+1);
} else if (auto a = is_neg(e)) {
if (auto r = to_num(*a, false))
return some(neg(*r));
}
return optional<mpz>();
}

2
src/library/num.h
View file

@ -12,6 +12,8 @@ namespace lean {
zero, one, bit0, bit1 */
bool has_num_decls(environment const & env);
bool is_const_app(expr const &, name const &, unsigned);
/** \brief Return true iff the given expression encodes a numeral. */
bool is_num(expr const & e);

3
src/library/tactic/norm_num_tactic.cpp
View file

@ -28,10 +28,10 @@ tactic norm_num_tactic() {
type_checker_ptr rtc = mk_type_checker(env, UnfoldReducible);
lhs = normalize(*rtc, lhs);
rhs = normalize(*rtc, rhs);
buffer<expr> hyps;
g.get_hyps(hyps);
local_context ctx(to_list(hyps));
// std::cout << "num of lhs: " << num_of_expr(env, ctx, lhs) << "\n";
try {
pair<expr, expr> p = mk_norm_num(env, ctx, lhs);
expr new_lhs = p.first;
@ -53,6 +53,7 @@ tactic norm_num_tactic() {
return none_proof_state();
}
} else {
std::cout << "lhs: " << new_lhs << ", rhs: " << new_rhs << "\n";
throw_tactic_exception_if_enabled(s, "norm_num tactic failed, one side is not a numeral");
return none_proof_state();
}

6
tests/lean/extra/num_norm1.lean
View file

@ -1,4 +1,4 @@
import algebra.numeral algebra.field
import algebra.numeral algebra.field data.nat
open algebra
variable {A : Type}
@ -29,6 +29,10 @@ example : (12 : A) = 0 + (2 + 3) + 7 := by norm_num
example : (105 : A) = 70 + (33 + 2) := by norm_num
example : (45000000000 : A) = 23000000000 + 22000000000 := by norm_num
example : (12 : A) - 4 - (5 + -2) = 5 := by norm_num
example : (12 : A) - 4 - (5 + -2) - 20 = -15 := by norm_num
exit
example : (0 : A) * 0 = 0 := by norm_num
example : (0 : A) * 1 = 0 := by norm_num
example : (0 : A) * 2 = 0 := by norm_num