From 4bf0519843c2a43b56eb25dd006f37ded9dbf230 Mon Sep 17 00:00:00 2001 From: Rob Lewis Date: Thu, 5 Nov 2015 17:33:53 -0500 Subject: [PATCH] feat(norm_num): numeral normalizer works for +, -, *, / --- library/algebra/field.lean | 8 + library/algebra/numeral.lean | 75 ++- src/library/norm_num.cpp | 622 +++++++++++++++++++------ src/library/norm_num.h | 34 ++ src/library/tactic/norm_num_tactic.cpp | 29 +- tests/lean/extra/num_norm1.lean | 32 +- 6 files changed, 638 insertions(+), 162 deletions(-) diff --git a/library/algebra/field.lean b/library/algebra/field.lean index b2c9cc1687..ed3ded3746 100644 --- a/library/algebra/field.lean +++ b/library/algebra/field.lean @@ -305,6 +305,13 @@ section field theorem field.div_mul_eq_div_mul_one_div (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) : a / (b * c) = (a / b) * (1 / c) := by rewrite [-!field.div_div_eq_div_mul Hb Hc, -div_eq_mul_one_div] + + theorem eq_of_mul_eq_mul_of_nonzero_left {a b c : A} (H : a ≠ 0) (H2 : a * b = a * c) : b = c := + by rewrite [-one_mul b, -div_self H, div_mul_eq_mul_div, H2, mul_div_cancel_left H] + + theorem eq_of_mul_eq_mul_of_nonzero_right {a b c : A} (H : c ≠ 0) (H2 : a * c = b * c) : a = b := + by rewrite [-mul_one a, -div_self H, -mul_div_assoc, H2, mul_div_cancel _ H] + end field structure discrete_field [class] (A : Type) extends field A := @@ -459,6 +466,7 @@ section discrete_field variable (a) theorem div_mul_eq_div_mul_one_div : a / (b * c) = (a / b) * (1 / c) := by rewrite [-div_div_eq_div_mul, -div_eq_mul_one_div] + end discrete_field end algebra diff --git a/library/algebra/numeral.lean b/library/algebra/numeral.lean index 1a3db70a07..ea54b4f2cc 100644 --- a/library/algebra/numeral.lean +++ b/library/algebra/numeral.lean @@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Author: Robert Y. Lewis -/ -import algebra.ring +import algebra.ordered_field open algebra variable {A : Type} @@ -140,20 +140,12 @@ 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 @@ -176,3 +168,68 @@ theorem subst_into_subtr [s : add_group A] (l r t : A) (prt : l + -r = t) : l - theorem neg_neg_helper [s : add_group A] (a b : A) (H : a = -b) : -a = b := by rewrite [H, neg_neg] + +theorem neg_mul_neg_helper [s : ring A] (a b c : A) (H : a * b = c) : (-a) * (-b) = c := + begin rewrite [neg_mul_neg, H] end + +theorem neg_mul_pos_helper [s : ring A] (a b c : A) (H : a * b = c) : (-a) * b = -c := + begin rewrite [-neg_mul_eq_neg_mul, H] end + +theorem pos_mul_neg_helper [s : ring A] (a b c : A) (H : a * b = c) : a * (-b) = -c := + begin rewrite [-neg_mul_comm, -neg_mul_eq_neg_mul, H] end + +theorem pos_bit0_helper [s : linear_ordered_semiring A] (a : A) (H : a > 0) : bit0 a > 0 := + by rewrite ↑bit0; apply add_pos H H + +theorem nonneg_bit0_helper [s : linear_ordered_semiring A] (a : A) (H : a ≥ 0) : bit0 a ≥ 0 := + by rewrite ↑bit0; apply add_nonneg H H + +theorem pos_bit1_helper [s : linear_ordered_semiring A] (a : A) (H : a ≥ 0) : bit1 a > 0 := + begin rewrite ↑bit1, apply add_pos_of_nonneg_of_pos, apply nonneg_bit0_helper _ H, apply zero_lt_one end + +theorem nonneg_bit1_helper [s : linear_ordered_semiring A] (a : A) (H : a ≥ 0) : bit1 a ≥ 0 := + by apply le_of_lt; apply pos_bit1_helper _ H + +theorem div_add_helper [s : field A] (n d b c val : A) (Hd : d ≠ 0) (H : n + b * d = val) (H2 : c * d = val) : + n / d + b = c := + begin + apply eq_of_mul_eq_mul_of_nonzero_right Hd, + rewrite [H2, -H, right_distrib, div_mul_cancel _ Hd] + end + +theorem add_div_helper [s : field A] (n d b c val : A) (Hd : d ≠ 0) (H : d * b + n = val) (H2 : d * c = val) : + b + n / d = c := + begin + apply eq_of_mul_eq_mul_of_nonzero_left Hd, + rewrite [H2, -H, left_distrib, mul_div_cancel' Hd] + end + +theorem div_mul_helper [s : field A] (n d c v : A) (Hd : d ≠ 0) (H : (n * c) / d = v) : (n / d) * c = v := + begin rewrite [-H, field.div_mul_eq_mul_div_comm _ _ Hd, mul_div_assoc] end + +theorem mul_div_helper [s : field A] (a n d v : A) (Hd : d ≠ 0) (H : (a * n) / d = v) : a * (n / d) = v := + begin rewrite [-H, mul_div_assoc] end + +theorem nonzero_of_pos_helper [s : linear_ordered_semiring A] (a : A) (H : a > 0) : a ≠ 0 := + ne_of_gt H + +theorem nonzero_of_neg_helper [s : linear_ordered_ring A] (a : A) (H : a > 0) : -a ≠ 0 := + begin apply ne_of_lt, apply neg_neg_of_pos H end + +theorem nonzero_of_div_helper [s : field A] (a b : A) (Ha : a ≠ 0) (Hb : b ≠ 0) : a / b ≠ 0 := + begin + intro Hab, + have Habb : (a / b) * b = 0, by rewrite [Hab, zero_mul], + rewrite [div_mul_cancel _ Hb at Habb], + exact Ha Habb + end + +theorem div_helper [s : field A] (n d v : A) (Hd : d ≠ 0) (H : v * d = n) : n / d = v := + begin apply eq_of_mul_eq_mul_of_nonzero_right Hd, rewrite (div_mul_cancel _ Hd), exact eq.symm H end + +theorem div_eq_div_helper [s : field A] (a b c d v : A) (H1 : a * d = v) (H2 : c * b = v) + (Hb : b ≠ 0) (Hd : d ≠ 0) : a / b = c / d := + begin apply eq_div_of_mul_eq, exact Hd, rewrite div_mul_eq_mul_div, apply eq.symm, apply eq_div_of_mul_eq, exact Hb, rewrite [H1, H2] end + +theorem subst_into_div [s : has_division A] (a₁ b₁ a₂ b₂ v : A) (H : a₁ / b₁ = v) (H1 : a₂ = a₁) (H2 : b₂ = b₁) : + a₂ / b₂ = v := by rewrite [H1, H2, H] diff --git a/src/library/norm_num.cpp b/src/library/norm_num.cpp index b5fe34a7c4..204896b0cb 100644 --- a/src/library/norm_num.cpp +++ b/src/library/norm_num.cpp @@ -13,6 +13,7 @@ static name * g_add = nullptr, * g_mul = nullptr, * g_sub = nullptr, * g_neg = nullptr, + * g_div = nullptr, * g_bit0_add_bit0 = nullptr, * g_bit1_add_bit0 = nullptr, * g_bit0_add_bit1 = nullptr, @@ -42,12 +43,14 @@ static name * g_add = nullptr, * g_has_mul = nullptr, * g_add_monoid = nullptr, * g_monoid = nullptr, + * g_ring = nullptr, * g_add_comm = nullptr, * g_add_group = nullptr, * g_mul_zero_class = nullptr, * g_distrib = nullptr, * g_has_neg = nullptr, * g_has_sub = nullptr, + * g_has_div = nullptr, * g_semiring = nullptr, * g_eq_neg_of_add_eq_zero = nullptr, * g_neg_add_neg_eq = nullptr, @@ -55,8 +58,32 @@ static name * g_add = nullptr, * g_neg_add_pos2 = nullptr, * g_pos_add_neg = nullptr, * g_pos_add_pos = nullptr, + * g_neg_mul_neg = nullptr, + * g_pos_mul_neg = nullptr, + * g_neg_mul_pos = nullptr, * g_sub_eq_add_neg = nullptr, * g_neg_neg = nullptr, + * g_div_add = nullptr, + * g_add_div = nullptr, + * g_lin_ord_ring = nullptr, + * g_lin_ord_semiring = nullptr, + * g_wk_order = nullptr, + * g_bit0_nonneg = nullptr, + * g_bit1_nonneg = nullptr, + * g_zero_le_one = nullptr, + * g_le_refl = nullptr, + * g_bit0_pos = nullptr, + * g_bit1_pos = nullptr, + * g_zero_lt_one = nullptr, + * g_field = nullptr, + * g_nonzero_neg = nullptr, + * g_nonzero_pos = nullptr, + * g_div_mul = nullptr, + * g_mul_div = nullptr, + * g_div_helper = nullptr, + * g_div_eq_div_helper = nullptr, + * g_subst_div = nullptr, + * g_nonzero_div = nullptr, * g_add_comm_group = nullptr; static bool is_numeral_aux(expr const & e, bool is_first) { @@ -69,8 +96,10 @@ static bool is_numeral_aux(expr const & e, bool is_first) { return args.size() == 2; } else if (const_name(f) == get_zero_name()) { return is_first && args.size() == 2; - } else if (const_name(f) == get_bit1_name() || const_name(f) == get_bit0_name()) { + } else if (const_name(f) == get_bit0_name()) { return args.size() == 3 && is_numeral_aux(args[2], false); + } else if (const_name(f) == get_bit1_name()) { + return args.size() == 4 && is_numeral_aux(args[3], false); } return false; } @@ -79,10 +108,14 @@ bool norm_num_context::is_numeral(expr const & e) const { return is_numeral_aux(e, true); } -bool is_neg(expr const & e) { +bool norm_num_context::is_neg_app(expr const & e) const { return is_const_app(e, *g_neg, 3); } +bool norm_num_context::is_div(expr const & e) const { + return is_const_app(e, *g_div, 4); +} + /* Takes A : Type, and tries to synthesize has_add A. */ @@ -146,6 +179,16 @@ expr norm_num_context::mk_monoid(expr const & e) { } } +expr norm_num_context::mk_field(expr const & e) { + expr t = mk_app(mk_constant(*g_field, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize field instance"); + } +} + expr norm_num_context::mk_add_comm(expr const & e) { expr t = mk_app(mk_constant(*g_add_comm, m_lvls), e); optional inst = mk_class_instance(m_env, m_ctx, t); @@ -216,6 +259,16 @@ expr norm_num_context::mk_has_sub(expr const & e) { } } +expr norm_num_context::mk_has_div(expr const & e) { + expr t = mk_app(mk_constant(*g_has_div, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize has_div 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 inst = mk_class_instance(m_env, m_ctx, t); @@ -226,6 +279,46 @@ expr norm_num_context::mk_add_comm_group(expr const & e) { } } +expr norm_num_context::mk_ring(expr const & e) { + expr t = mk_app(mk_constant(*g_ring, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize ring instance"); + } +} + +expr norm_num_context::mk_lin_ord_ring(expr const & e) { + expr t = mk_app(mk_constant(*g_lin_ord_ring, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize lin_ord_ring instance"); + } +} + +expr norm_num_context::mk_lin_ord_semiring(expr const & e) { + expr t = mk_app(mk_constant(*g_lin_ord_semiring, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize lin_ord_semiring instance"); + } +} + +expr norm_num_context::mk_wk_order(expr const & e) { + expr t = mk_app(mk_constant(*g_wk_order, m_lvls), e); + optional inst = mk_class_instance(m_env, m_ctx, t); + if (inst) { + return *inst; + } else { + throw exception("failed to synthesize weak_order instance"); + } +} + expr norm_num_context::mk_const(name const & n) { return mk_constant(n, m_lvls); } @@ -234,46 +327,6 @@ 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 norm_num_context::mk_norm(expr const & e) { - buffer args; - expr f = get_app_args(e, args); - if (!is_constant(f)) { - throw exception("cannot take norm of nonconstant"); - } - m_lvls = const_levels(f); - if (const_name(f) == *g_add && args.size() == 4) { - 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[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(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[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(mul_p.first, prf); - } else if (const_name(f) == get_bit0_name() && args.size() == 3) { - auto arg = mk_norm(args[2]); - expr rv = mk_app({f, args[0], args[1], arg.first}); - expr prf = mk_cong(mk_app({f, args[0], args[1]}), args[0], args[2], arg.first, arg.second); - return pair(rv, prf); - } else if (const_name(f) == get_bit1_name() && args.size() == 4) { - auto arg = mk_norm(args[3]); - expr rv = mk_app({f, args[0], args[1], args[2], arg.first}); - expr prf = mk_cong(mk_app({f, args[0], args[1], args[2]}), args[0], args[3], arg.first, arg.second); - return pair(rv, prf); - } else if ((const_name(f) == get_zero_name() || const_name(f) == get_one_name()) && args.size() == 2) { - return pair(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 "); - } - // TODO(Rob): cases for sub, div - }*/ - // returns such that p is a proof that lhs + rhs = t. pair norm_num_context::mk_norm_add(expr const & lhs, expr const & rhs) { buffer args_lhs; @@ -332,7 +385,9 @@ pair norm_num_context::mk_norm_add(expr const & lhs, expr const & rh rv = 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"; + std::cout << "\n\n bad args: " << lhs_head << ", " << rhs_head << "\n"; + std::cout << "\nlhs: " << lhs; + std::cout << "\nrhs: " << rhs; throw exception("mk_norm_add got malformed args"); } return pair(rv, prf); @@ -395,6 +450,7 @@ pair norm_num_context::mk_norm_mul(expr const & lhs, expr const & rh 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]); + // std::cout << "** in mk_norm_mul. calling mk_norm_add on" << mtp.first << ", " << lhs; 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(type), lhs, args_rhs[3], @@ -416,7 +472,46 @@ pair 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 +optional norm_num_context::to_mpq(expr const & e) { + auto v = to_num(e); + if (v) { + return optional(mpq(*v)); + } else { + return optional(); + } +} + +mpq norm_num_context:: mpq_of_expr(expr const & e){ + buffer args; + expr f = get_app_args(e, args); + if (!is_constant(f)) { + throw exception("cannot find num of nonconstant"); + } else if (const_name(f) == *g_add && args.size() == 4) { + return mpq_of_expr(args[2]) + mpq_of_expr(args[3]); + } else if (const_name(f) == *g_mul && args.size() == 4) { + return mpq_of_expr(args[2]) * mpq_of_expr(args[3]); + } else if (const_name(f) == *g_sub && args.size() == 4) { + return mpq_of_expr(args[2]) - mpq_of_expr(args[3]); + } else if (const_name(f) == *g_div && args.size() == 4) { + mpq num = mpq_of_expr(args[2]), den = mpq_of_expr(args[3]); + if (den != 0) + return mpq_of_expr(args[2]) / mpq_of_expr(args[3]); + else + throw exception("divide by 0"); + } else if (const_name(f) == *g_neg && args.size() == 3) { + return neg(mpq_of_expr(args[2])); + } else { + auto v = to_mpq(e); + if (v) { + return *v; + } else { + std::cout << "error : " << f << args.size() << "\n"; + throw exception("expression in mpq_of_expr is malfomed"); + } + } +} + +mpz norm_num_context::num_of_expr(expr const & e) { buffer args; expr f = get_app_args(e, args); if (!is_constant(f)) { @@ -440,8 +535,8 @@ mpz norm_num_context::num_of_expr(expr const & e) { // note : mpz only supports } } -pair get_type_and_arg_of_neg(expr & e) { - lean_assert(is_neg(e)); +pair norm_num_context::get_type_and_arg_of_neg(expr & e) { + lean_assert(is_neg_app(e)); buffer args; expr f = get_app_args(e, args); return pair(args[0], args[2]); @@ -449,9 +544,9 @@ pair get_type_and_arg_of_neg(expr & e) { // 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)); + lean_assert(is_neg_app(s_lhs)); + lean_assert(is_neg_app(s_rhs)); + lean_assert(is_neg_app(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); @@ -461,11 +556,11 @@ expr norm_num_context::mk_norm_eq_neg_add_neg(expr & s_lhs, expr & s_rhs, expr & } 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)); + lean_assert(is_neg_app(s_lhs)); + lean_assert(!is_neg_app(s_rhs)); auto s_lhs_v = get_type_and_arg_of_neg(s_lhs); expr type = s_lhs_v.first; - if (is_neg(rhs)) { + if (is_neg_app(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}); @@ -475,8 +570,9 @@ expr norm_num_context::mk_norm_eq_neg_add_pos(expr & s_lhs, expr & s_rhs, expr & } } - expr norm_num_context::mk_norm_eq_pos_add_neg(expr & s_lhs, expr & s_rhs, expr & rhs) { + lean_assert(is_neg_app(s_rhs)); + lean_assert(!is_neg_app(s_lhs)); 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}); @@ -484,80 +580,57 @@ expr norm_num_context::mk_norm_eq_pos_add_neg(expr & s_lhs, expr & s_rhs, expr & // 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)); + lean_assert(!is_neg_app(s_lhs)); + lean_assert(!is_neg_app(s_rhs)); + lean_assert(!is_neg_app(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_neg_mul_neg(expr & s_lhs, expr & s_rhs, expr & rhs) { + lean_assert(is_neg_app(s_lhs)); + lean_assert(is_neg_app(s_rhs)); + lean_assert(is_neg_app(rhs)); + auto s_lhs_v = get_type_and_arg_of_neg(s_lhs).second; + expr s_rhs_v, type; + std::tie(type, s_rhs_v) = get_type_and_arg_of_neg(s_rhs); + auto prod_pr = mk_norm(mk_mul(type, s_lhs_v, s_rhs_v)); //, rhs); + lean_assert(to_num(rhs) == to_num(prod_pr.first)); + return mk_app({mk_const(*g_neg_mul_neg), type, mk_ring(type), s_lhs_v, s_rhs_v, rhs, prod_pr.second}); +} -/*expr norm_num_context::mk_norm_eq(expr const & lhs, expr const & rhs) { // rhs is a nonneg numeral - buffer 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 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 rvargs = buffer(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::mk_norm_eq_neg_mul_pos(expr & s_lhs, expr & s_rhs, expr & rhs) { + lean_assert(is_neg_app(s_lhs)); + lean_assert(!is_neg_app(s_rhs)); + lean_assert(is_neg_app(rhs)); + expr s_lhs_v, type; + std::tie(type, s_lhs_v) = get_type_and_arg_of_neg(s_lhs); + auto rhs_v = get_type_and_arg_of_neg(rhs).second; + auto prod_pr = mk_norm(mk_mul(type, s_lhs_v, s_rhs)); //, rhs_v); + return mk_app({mk_const(*g_neg_mul_pos), type, mk_ring(type), s_lhs_v, s_rhs, rhs_v, prod_pr.second}); +} + +expr norm_num_context::mk_norm_eq_pos_mul_neg(expr & s_lhs, expr & s_rhs, expr & rhs) { + lean_assert(!is_neg_app(s_lhs)); + lean_assert(is_neg_app(s_rhs)); + lean_assert(is_neg_app(rhs)); + expr s_rhs_v, type; + std::tie(type, s_rhs_v) = get_type_and_arg_of_neg(s_rhs); + auto rhs_v = get_type_and_arg_of_neg(rhs).second; + auto prod_pr = mk_norm(mk_mul(type, s_lhs, s_rhs_v)); //, rhs_v); + return mk_app({mk_const(*g_pos_mul_neg), type, mk_ring(type), s_lhs, s_rhs_v, rhs_v, prod_pr.second}); +} + +// returns a proof that s_lhs + s_rhs = rhs, where all are nonneg normalized numerals +expr norm_num_context::mk_norm_eq_pos_mul_pos(expr & s_lhs, expr & s_rhs, expr & rhs) { + lean_assert(!is_neg_app(s_lhs)); + lean_assert(!is_neg_app(s_rhs)); + lean_assert(!is_neg_app(rhs)); + auto p = mk_norm_mul(s_lhs, s_rhs); + lean_assert(to_num(rhs) == to_num(p.first)); + return p.second; +} expr norm_num_context::from_pos_num(mpz const & n, expr const & type) { lean_assert(n > 0); @@ -580,6 +653,23 @@ expr norm_num_context::from_num(mpz const & n, expr const & type) { return r; } +// assumes q >= 0 +expr norm_num_context::from_mpq(mpq const & q, expr const & type) { + mpz numer = q.get_numerator(); + mpz denom = q.get_denominator(); + lean_assert(numer >= 0 && denom >= 0); + if (denom == 1) { + return from_num(numer, type); + } else { + return mk_div(type, from_num(numer, type), from_num(denom, type)); + } +} + +expr norm_num_context::mk_div(expr const & type, expr const & e1, expr const & e2) { + auto has_div = mk_has_div(type); + return mk_app({mk_const(*g_div), type, has_div, e1, e2}); +} + 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}); @@ -590,8 +680,157 @@ expr norm_num_context::mk_add(expr const & type, expr const & e1, expr const & e return mk_app({mk_const(*g_add), type, has_add, e1, e2}); } +expr norm_num_context::mk_mul(expr const & type, expr const & e1, expr const & e2) { + auto has_mul = mk_has_mul(type); + return mk_app({mk_const(*g_mul), type, has_mul, e1, e2}); +} + +/*mpz mpz_gcd(mpz a, mpz b) { + return b == 0 ? a : mpz_gcd(b, a % b); +}*/ + +/*pair norm_num_context::mk_norm_div_over_div(expr & lhs, expr & rhs) { + + buffer lhs_args, rhs_args; + get_app_args(lhs, lhs_args); + get_app_args(rhs, rhs_args); + expr type = lhs_args[0]; + expr a = lhs_args[2], b = lhs_args[3], c = rhs_args[2], d = rhs_args[3]; + lean_assert(!is_div(a) && !is_div(b) && !is_div(c) && !is_div(d)); + // normalizing (a/b) / (c/d), where a, b, c, d are not divs + auto nlhs = mk_norm(mk_mul(type, a, d)); + auto nrhs = mk_norm(mk_mul(type, b, c)); + auto nlhs_val = to_num(nlhs.first), rlhs_val = to_num(nrhs.first); + if (nlhs_val && rlhs_val) { + + } +} + +pair norm_num_context::mk_norm_div_over_num(expr &, expr &) { +} + +pair norm_num_context::mk_norm_num_over_div(expr &, expr &) { + +} + +pair norm_num_context::mk_norm_num_over_num(expr &, expr &) { + +}*/ + +// s_lhs is div. returns proof that s_lhs + s_rhs = rhs +expr norm_num_context::mk_norm_div_add(expr & s_lhs, expr & s_rhs, expr & rhs) { + buffer s_lhs_args; + get_app_args(s_lhs, s_lhs_args); + expr type = s_lhs_args[0]; + expr num = s_lhs_args[2], den = s_lhs_args[3]; + expr new_lhs = mk_add(type, num, mk_mul(type, s_rhs, den)); + auto npr_l = mk_norm(new_lhs); + auto npr_r = mk_norm(mk_mul(type, rhs, den)); + lean_assert(to_mpq(npr_l.first) == to_mpq(npr_r.first)); + if (!(is_numeral(den))) { std::cout << "\n****bad input in mk_norm_div_add\n"; } + expr den_neq_zero = mk_nonzero_prf(den); + return mk_app({mk_const(*g_div_add), type, mk_field(type), num, den, s_rhs, rhs, + npr_l.first, den_neq_zero, npr_l.second, npr_r.second}); +} + +// s_rhs is div. returns proof that s_lhs + s_rhs = rhs +expr norm_num_context::mk_norm_add_div(expr & s_lhs, expr & s_rhs, expr & rhs) { + buffer s_rhs_args; + get_app_args(s_rhs, s_rhs_args); + expr type = s_rhs_args[0]; + expr num = s_rhs_args[2], den = s_rhs_args[3]; + expr new_lhs = mk_add(type, mk_mul(type, den, s_lhs), num); + auto npr_l = mk_norm(new_lhs); + auto npr_r = mk_norm(mk_mul(type, den, rhs)); + lean_assert(to_mpq(npr_l.first) == to_mpq(npr_r.first)); + if (!(is_numeral(den))) { std::cout << "\n****bad input in mk_norm_add_div\n"; } + expr den_neq_zero = mk_nonzero_prf(den); + return mk_app({mk_const(*g_add_div), type, mk_field(type), num, den, s_rhs, rhs, + npr_l.first, den_neq_zero, npr_l.second, npr_r.second}); +} + +// if e is a numeral or a negation of a numeral or division, returns proof that e != 0 +expr norm_num_context::mk_nonzero_prf(expr const & e) { + buffer args; + expr f = get_app_args(e, args); + if (const_name(f) == *g_neg) { + return mk_app({mk_const(*g_nonzero_neg), args[0], mk_lin_ord_ring(args[0]), args[2], mk_pos_prf(args[2])}); + } else if (const_name(f) == *g_div) { + expr num_pr = mk_nonzero_prf(args[2]), den_pr = mk_nonzero_prf(args[3]); + return mk_app({mk_const(*g_nonzero_div), args[0], mk_field(args[0]), args[2], args[3], num_pr, den_pr}); + } else { + return mk_app({mk_const(*g_nonzero_pos), args[0], mk_lin_ord_semiring(args[0]), e, mk_pos_prf(e)}); + } +} + +// if e is a numeral, makes a proof that e > 0 +expr norm_num_context::mk_pos_prf(expr const & e) { + buffer args; + get_app_args(e, args); + expr type = args[0]; + expr prf; + if (is_bit0(e)) { + prf = mk_pos_prf(args[2]); + return mk_app({mk_const(*g_bit0_pos), type, mk_lin_ord_semiring(type), args[2], prf}); + } else if (is_bit1(e)) { + prf = mk_nonneg_prf(args[3]); + return mk_app({mk_const(*g_bit1_pos), type, mk_lin_ord_semiring(type), args[3], prf}); + } else if (is_one(e)) { + return mk_app({mk_const(*g_zero_lt_one), type, mk_lin_ord_semiring(type)}); + } else { + std::cout << "bad call to mk_pos_prf: " << e << "\n"; + throw exception("mk_pos_proof called on zero or non_numeral"); + } +} + +expr norm_num_context::mk_nonneg_prf(expr const & e) { + buffer args; + get_app_args(e, args); + expr type = args[0]; + expr prf; + if (is_bit0(e)) { + prf = mk_nonneg_prf(args[2]); + return mk_app({mk_const(*g_bit0_nonneg), type, mk_lin_ord_semiring(type), args[2], prf}); + } else if (is_bit1(e)) { + prf = mk_nonneg_prf(args[3]); + return mk_app({mk_const(*g_bit1_nonneg), type, mk_lin_ord_semiring(type), args[3], prf}); + } else if (is_one(e)) { + return mk_app({mk_const(*g_zero_le_one), type, mk_lin_ord_ring(type)}); + } else if (is_zero(e)) { + return mk_app({mk_const(*g_le_refl), type, mk_wk_order(type), mk_app({mk_const(get_zero_name()), type, mk_has_zero(type)})}); + } else { + std::cout << "bad call to mk_nonneg_prf: " << e << "\n"; + throw exception("mk_nonneg_proof called on zero or non_numeral"); + } +} + +// s_lhs is div. returns proof that s_lhs * s_rhs = rhs +expr norm_num_context::mk_norm_div_mul(expr & s_lhs, expr & s_rhs, expr & rhs) { + buffer args; + get_app_args(s_lhs, args); + expr type = args[0]; + expr new_num = mk_mul(type, args[2], s_rhs); + auto prf = mk_norm(mk_div(type, new_num, args[3])); + lean_assert(to_mpq(prf.first) == to_mpq(rhs)); + if (!(is_numeral(args[3]))) { std::cout << "\n****bad input in mk_norm_div_mul\n"; } + expr den_ne_zero = mk_nonzero_prf(args[3]); + return mk_app({mk_const(*g_div_mul), type, mk_field(type), args[2], args[3], s_rhs, rhs, den_ne_zero, prf.second}); +} + +expr norm_num_context::mk_norm_mul_div(expr & s_lhs, expr & s_rhs, expr & rhs) { + buffer args; + get_app_args(s_rhs, args); + expr type = args[0]; + expr new_num = mk_mul(type, s_lhs, args[2]); + auto prf = mk_norm(mk_div(type, new_num, args[3])); + lean_assert(to_mpq(prf.first) == to_mpq(rhs)); + if (!(is_numeral(args[3]))) { std::cout << "\n****bad input in mk_norm_mul_div\n"; } + expr den_ne_zero = mk_nonzero_prf(args[3]); + return mk_app({mk_const(*g_mul_div), type, mk_field(type), s_lhs, args[2], args[3], rhs, den_ne_zero, prf.second}); +} + pair norm_num_context::mk_norm(expr const & e) { - std::cout << "mk_norm\n"; + //std::cout << "mk_norm: " << e << "\n"; buffer args; expr f = get_app_args(e, args); if (!is_constant(f) || args.size() == 0) { @@ -599,25 +838,29 @@ pair norm_num_context::mk_norm(expr const & e) { } m_lvls = const_levels(f); expr type = args[0]; - auto val = num_of_expr(e); + mpq val = mpq_of_expr(e); expr nval; // e = nval if (val >= 0) { - nval = from_num(val, type); + nval = from_mpq(val, type); } else { - nval = mk_neg(type, from_num(neg(val), type)); + nval = mk_neg(type, from_mpq(neg(val), type)); } if (const_name(f) == *g_add && args.size() == 4) { + expr prf; 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)) { + if (is_div(lhs_p.first)) { + prf = mk_norm_div_add(lhs_p.first, rhs_p.first, nval); + } else if (is_div(rhs_p.first)) { + prf = mk_norm_add_div(lhs_p.first, rhs_p.first, nval); + } else if (is_neg_app(lhs_p.first)) { + if (is_neg_app(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)) { + if (is_neg_app(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); @@ -634,16 +877,75 @@ pair norm_num_context::mk_norm(expr const & e) { return pair(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)) { + /*std::cout << "in mk_norm. const is neg. e: " << e << "\n"; + std::cout << " val: "; + std::cout << val; + std::cout << "\n nval: "; + std::cout << nval << "\n";*/ + lean_assert(mpq_of_expr(prf.first) == neg(val)); + + if (is_neg_app(nval)) { buffer 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); + expr rprf = mk_cong(mk_app(f, args[0], args[1]), type, args[2], nval_args[2], prf.second); + // std::cout << " RETURNING: (" << nval << ", " << rprf << "\n"; return pair(nval, rprf); } else { - auto rprf = mk_app({mk_const(*g_neg_neg), type, mk_add_group(type), args[2], nval, prf.second}); + expr rprf = mk_app({mk_const(*g_neg_neg), type, mk_add_group(type), args[2], nval, prf.second}); return pair(nval, rprf); } + } else if (const_name(f) == *g_mul && args.size() == 4) { + auto lhs_p = mk_norm(args[2]); + auto rhs_p = mk_norm(args[3]); + expr prf; + if (is_div(lhs_p.first)) { + prf = mk_norm_div_mul(lhs_p.first, rhs_p.first, nval); + } else if (is_div(rhs_p.first)) { + prf = mk_norm_mul_div(lhs_p.first, rhs_p.first, nval); + } else if (is_zero(lhs_p.first) || is_zero(rhs_p.first)) { + prf = mk_norm_mul(lhs_p.first, rhs_p.first).second; + } else if (is_neg_app(lhs_p.first)) { + if (is_neg_app(rhs_p.first)) { + prf = mk_norm_eq_neg_mul_neg(lhs_p.first, rhs_p.first, nval); + } else { // bad args passing here + prf = mk_norm_eq_neg_mul_pos(lhs_p.first, rhs_p.first, nval); + } + } else { + if (is_neg_app(rhs_p.first)) { + prf = mk_norm_eq_pos_mul_neg(lhs_p.first, rhs_p.first, nval); + } else { + prf = mk_norm_eq_pos_mul_pos(lhs_p.first, rhs_p.first, nval); + } + } + expr rprf = mk_app({mk_const(*g_subst_prod), type, mk_has_mul(args[0]), args[2], args[3], + lhs_p.first, rhs_p.first, nval, lhs_p.second, rhs_p.second, prf}); + return pair(nval, rprf); + } else if (const_name(f) == get_division_name() && args.size() == 4) { + auto lhs_p = mk_norm(args[2]); + auto rhs_p = mk_norm(args[3]); + //std::cout << "div. lhs, rhs:" << lhs_p.first << ",\n" << rhs_p.first << ".\n"; + expr prf; + if (is_div(nval)) { + // std::cout << "nval is div. (" << nval << ")\n"; + buffer nval_args; + get_app_args(nval, nval_args); + expr nval_num = nval_args[2], nval_den = nval_args[3]; + auto lhs_mul = mk_norm(mk_mul(type, lhs_p.first, nval_den)); + auto rhs_mul = mk_norm(mk_mul(type, nval_num, rhs_p.first)); + expr den_nonzero = mk_nonzero_prf(rhs_p.first); + expr nval_den_nonzero = mk_nonzero_prf(nval_den); + prf = mk_app({mk_const(*g_div_eq_div_helper), type, mk_field(type), lhs_p.first, rhs_p.first, nval_num, nval_den, lhs_mul.first, lhs_mul.second, rhs_mul.second, den_nonzero, nval_den_nonzero}); + } else { + auto prod = mk_norm(mk_mul(type, nval, rhs_p.first)); + auto val1 = to_mpq(prod.first), val2 = to_mpq(lhs_p.first); + if (val1 && val2) { + lean_assert(*val1 == *val2); + } + expr den_nonzero = mk_nonzero_prf(rhs_p.first); + prf = mk_app({mk_const(*g_div_helper), type, mk_field(type), lhs_p.first, rhs_p.first, nval, den_nonzero, prod.second}); + } + expr rprf = mk_app({mk_const(*g_subst_div), type, mk_has_div(type), lhs_p.first, rhs_p.first, args[2], args[3], nval, prf, lhs_p.second, rhs_p.second}); + return pair(nval, rprf); } else if (const_name(f) == get_bit0_name() && args.size() == 3) { lean_assert(is_bit0(nval)); buffer nval_args; @@ -659,7 +961,9 @@ pair norm_num_context::mk_norm(expr const & e) { auto rprf = mk_cong(mk_app(f, args[0], args[1], args[2]), type, args[3], nval_args[3], prf.second); return pair(nval, rprf); } else if ((const_name(f) == get_zero_name() || const_name(f) == get_one_name()) && args.size() == 2) { - return pair(e, mk_app({mk_const(*g_mk_eq), args[0], e})); + auto p = pair(e, mk_app({mk_const(*g_mk_eq), args[0], e})); +// return pair(e, mk_app({mk_const(*g_mk_eq), args[0], e})); + return p; } else { std::cout << "error with name " << const_name(f) << " and size " << args.size() << ".\n"; throw exception("mk_norm found unrecognized combo "); @@ -672,6 +976,7 @@ void initialize_norm_num() { g_mul = new name("mul"); g_sub = new name("sub"); g_neg = new name("neg"); + g_div = new name("division"); 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"); @@ -700,23 +1005,49 @@ void initialize_norm_num() { g_mul_bit1 = new name("mul_bit1_helper"); g_has_mul = new name("has_mul"); g_add_monoid = new name("algebra", "add_monoid"); + g_ring = new name("algebra", "ring"); 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_has_sub = new name("has_sub"); + g_has_div = new name("has_division"); g_semiring = new name("algebra", "semiring"); + g_lin_ord_ring = new name("algebra", "linear_ordered_ring"); + g_lin_ord_semiring = new name("algebra", "linear_ordered_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_neg_mul_neg = new name("neg_mul_neg_helper"); + g_neg_mul_pos = new name("neg_mul_pos_helper"); + g_pos_mul_neg = new name("pos_mul_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"); + g_add_div = new name("add_div_helper"); + g_div_add = new name("div_add_helper"); + g_bit0_nonneg = new name("nonneg_bit0_helper"); + g_bit1_nonneg = new name("nonneg_bit1_helper"); + g_zero_le_one = new name("algebra", "zero_le_one"); + g_le_refl = new name("algebra", "le.refl"); + g_bit0_pos = new name("pos_bit0_helper"); + g_bit1_pos = new name("pos_bit1_helper"); + g_zero_lt_one = new name("algebra", "zero_lt_one"); + g_wk_order = new name("algebra", "weak_order"); + g_field = new name("algebra", "field"); + g_nonzero_neg = new name("nonzero_of_neg_helper"); + g_nonzero_pos = new name("nonzero_of_pos_helper"); + g_mul_div = new name("mul_div_helper"); + g_div_mul = new name("div_mul_helper"); + g_div_helper = new name("div_helper"); + g_div_eq_div_helper = new name("div_eq_div_helper"); + g_subst_div = new name("subst_into_div"); + g_nonzero_div = new name("nonzero_of_div_helper"); } void finalize_norm_num() { @@ -725,6 +1056,7 @@ void finalize_norm_num() { delete g_mul; delete g_sub; delete g_neg; + delete g_div; delete g_bit0_add_bit0; delete g_bit1_add_bit0; delete g_bit0_add_bit1; @@ -754,12 +1086,14 @@ void finalize_norm_num() { delete g_has_mul; delete g_add_monoid; delete g_monoid; + delete g_ring; 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_has_div; delete g_semiring; delete g_eq_neg_of_add_eq_zero; delete g_neg_add_neg_eq; @@ -767,8 +1101,26 @@ void finalize_norm_num() { delete g_neg_add_pos2; delete g_pos_add_neg; delete g_pos_add_pos; + delete g_neg_mul_neg; + delete g_neg_mul_pos; + delete g_pos_mul_neg; delete g_sub_eq_add_neg; delete g_neg_neg; delete g_add_comm_group; + delete g_div_add; + delete g_add_div; + delete g_bit0_nonneg; + delete g_bit1_nonneg; + delete g_zero_le_one; + delete g_le_refl; + delete g_bit0_pos; + delete g_bit1_pos; + delete g_zero_lt_one; + delete g_wk_order; + delete g_div_mul; + delete g_div_helper; + delete g_div_eq_div_helper; + delete g_mul_div; + delete g_nonzero_div; } } diff --git a/src/library/norm_num.h b/src/library/norm_num.h index b528eaf1b4..c6d03d9a32 100644 --- a/src/library/norm_num.h +++ b/src/library/norm_num.h @@ -8,6 +8,7 @@ Author: Robert Y. Lewis #include "library/local_context.h" #include "library/num.h" #include "library/class_instance_resolution.h" +#include "util/numerics/mpq.h" namespace lean { class norm_num_context { @@ -31,27 +32,56 @@ class norm_num_context { expr mk_add_comm(expr const &); expr mk_add_group(expr const &); expr mk_mul_zero_class(expr const &); + expr mk_ring(expr const &); expr mk_semiring(expr const &); + expr mk_field(expr const &); + expr mk_lin_ord_semiring(expr const &); + expr mk_lin_ord_ring(expr const &); + expr mk_wk_order(expr const &); expr mk_has_neg(expr const &); + expr mk_has_div(expr const &); expr mk_has_sub(expr const &); expr mk_add(expr const &, expr const &, expr const &); + expr mk_mul(expr const &, expr const &, expr const &); + expr mk_div(expr const &, expr const &, expr const &); expr mk_neg(expr const &, expr const &); expr mk_add_comm_group(expr const &); + expr mk_pos_prf(expr const &); + expr mk_nonneg_prf(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 &); + expr mk_norm_eq_neg_mul_neg(expr &, expr &, expr &); + expr mk_norm_eq_neg_mul_pos(expr &, expr &, expr &); + expr mk_norm_eq_pos_mul_neg(expr &, expr &, expr &); + expr mk_norm_eq_pos_mul_pos(expr &, expr &, expr &); + //pair mk_norm_div_over_div(expr &, expr &); + //pair mk_norm_div_over_num(expr &, expr &); + //pair mk_norm_num_over_div(expr &, expr &); + //pair mk_norm_num_over_num(expr &, expr &); + expr mk_norm_div_add(expr &, expr &, expr &); + expr mk_norm_add_div(expr &, expr &, expr &); + expr mk_norm_div_mul(expr &, expr &, expr &); + expr mk_norm_mul_div(expr &, expr &, expr &); + expr mk_nonzero_prf(expr const & e); + pair get_type_and_arg_of_neg(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; + bool is_neg_app(expr const &) const; + bool is_div(expr const &) const; pair mk_norm(expr const & e); //pair mk_norm_expr(expr const & e); expr mk_norm_eq(expr const &, expr const &); mpz num_of_expr(expr const & e); + mpq mpq_of_expr(expr const & e); + optional to_mpq(expr const & e); expr from_pos_num(mpz const &, expr const &); expr from_num(mpz const &, expr const &); + expr from_mpq(mpq const &, expr const &); }; inline bool is_neg(expr const & e); @@ -68,6 +98,10 @@ inline mpz num_of_expr(environment const & env, local_context const & ctx, expr return norm_num_context(env, ctx).num_of_expr(e); } +inline mpq mpq_of_expr(environment const & env, local_context const & ctx, expr const & e) { + return norm_num_context(env, ctx).mpq_of_expr(e); +} + void initialize_norm_num(); void finalize_norm_num(); } diff --git a/src/library/tactic/norm_num_tactic.cpp b/src/library/tactic/norm_num_tactic.cpp index 461b515e32..6ea3a5d5c0 100644 --- a/src/library/tactic/norm_num_tactic.cpp +++ b/src/library/tactic/norm_num_tactic.cpp @@ -31,32 +31,31 @@ tactic norm_num_tactic() { buffer hyps; g.get_hyps(hyps); local_context ctx(to_list(hyps)); -// std::cout << "num of lhs: " << num_of_expr(env, ctx, lhs) << "\n"; + //std::cout << "\nnum of lhs: " << mpq_of_expr(env, ctx, lhs) << "\n"; try { pair p = mk_norm_num(env, ctx, lhs); + //std::cout << "checkpt 0"; expr new_lhs = p.first; expr new_lhs_pr = p.second; pair p2 = mk_norm_num(env, ctx, rhs); expr new_rhs = p2.first; expr new_rhs_pr = p2.second; - auto v_lhs = to_num(new_lhs), v_rhs = to_num(new_rhs); - if (v_lhs && v_rhs) { - if (*v_lhs == *v_rhs) { - type_checker tc(env); - expr g_prf = mk_trans(tc, new_lhs_pr, mk_symm(tc, new_rhs_pr)); - substitution new_subst = s.get_subst(); - assign(new_subst, g, g_prf); - return some_proof_state(proof_state(s, tail(gs), new_subst)); - } else { - std::cout << "lhs: " << new_lhs << ", rhs: " << new_rhs << "\n"; - throw_tactic_exception_if_enabled(s, "norm_num tactic failed, lhs doesn't match rhs"); - return none_proof_state(); - } + mpq v_lhs = mpq_of_expr(env, ctx, new_lhs), v_rhs = mpq_of_expr(env, ctx, new_rhs); + if (v_lhs == v_rhs) { + // std::cout << "checkpt 1\n"; + type_checker tc(env); + //std::cout << "checkpt 2: " << new_lhs_pr << ", \n" << new_rhs_pr << "\n"; + expr g_prf = mk_trans(tc, new_lhs_pr, mk_symm(tc, new_rhs_pr)); + //std::cout << "checkpt 3\n"; + substitution new_subst = s.get_subst(); + assign(new_subst, g, g_prf); + return some_proof_state(proof_state(s, tail(gs), new_subst)); } 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"); + throw_tactic_exception_if_enabled(s, "norm_num tactic failed, lhs doesn't match rhs"); return none_proof_state(); } + } catch (exception & ex) { throw_tactic_exception_if_enabled(s, ex.what()); return none_proof_state(); diff --git a/tests/lean/extra/num_norm1.lean b/tests/lean/extra/num_norm1.lean index 6942f37b21..dad01ebc32 100644 --- a/tests/lean/extra/num_norm1.lean +++ b/tests/lean/extra/num_norm1.lean @@ -2,9 +2,17 @@ import algebra.numeral algebra.field data.nat open algebra variable {A : Type} -variable [s : comm_ring A] +variable [s : linear_ordered_field A] include s +example : (-1 :A) * 1 = -1 := by norm_num +example : (-2 :A) * 1 = -2 := by norm_num +example : (-2 :A) * -1 = 2 := by norm_num +example : (-2 :A) * -2 = 4 := by norm_num +example : (1 : A) * 0 = 0 := by norm_num + +example : ((1 : A) + 1) * 5 = 6 + 4 := by norm_num + example : (1 : A) = 0 + 1 := by norm_num example : (1 : A) = 1 + 0 := by norm_num example : (2 : A) = 1 + 1 := by norm_num @@ -27,11 +35,17 @@ example : 33 = 5 + (28 : A) := by norm_num 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 +theorem name : (45000000000 : A) = 23000000000 + 22000000000 := by norm_num + +example : (0 : A) - 3 = -3 := by norm_num +example : (0 : A) - 2 = -2 := by norm_num +example : (1 : A) - 3 = -2 := by norm_num +example : (1 : A) - 1 = 0 := by norm_num +example : (0 : A) - 3 = -3 := by norm_num +example : (0 : A) - 3 = -3 := 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 @@ -52,3 +66,15 @@ example : (4 : A) * 4 = 16 := by norm_num example : (11 : A) * 2 = 22 := by norm_num example : (15 : A) * 6 = 90 := by norm_num example : (123456 : A) * 123456 = 15241383936 := by norm_num + +example : (4 : A) / 2 = 2 := by norm_num +example : (4 : A) / 1 = 4 := by norm_num +example : (4 : A) / 3 = 4 / 3 := by norm_num +example : (50 : A) / 5 = 10 := by norm_num +example : (1056 : A) / 1 = 1056 := by norm_num +example : (6 : A) / 4 = 3/2 := by norm_num +example : (0 : A) / 3 = 0 := by norm_num +--example : (3 : A) / 0 = 0 := by norm_num -- this should fail +example : (9 * 9 * 9) * (12 : A) / 27 = 81 * (2 + 2) := by norm_num +example : (-2 : A) * 4 / 3 = -8 / 3 := by norm_num +example : - (-4 / 3) = 1 / (3 / (4 : A)) := by norm_num