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feat(library/defitional): add no_confusion_type construction for inductive datatypes that are not propositions

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This commit is contained in:
Leonardo de Moura 2014-11-08 15:20:19 -08:00
parent 46149d0d50
commit b5da143fc0
15 changed files with 240 additions and 76 deletions
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13
src/frontends/lean/inductive_cmd.cpp
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@ -24,6 +24,9 @@ Author: Leonardo de Moura
#include "library/definitional/rec_on.h"
#include "library/definitional/induction_on.h"
#include "library/definitional/cases_on.h"
#include "library/definitional/no_confusion.h"
#include "library/definitional/eq.h"
#include "library/definitional/heq.h"
#include "library/definitional/unit.h"
#include "frontends/lean/decl_cmds.h"
#include "frontends/lean/util.h"
@ -637,6 +640,8 @@ struct inductive_cmd_fn {
environment mk_aux_decls(environment env, buffer<inductive_decl> const & decls) {
bool has_unit = has_unit_decls(env);
bool has_eq = has_eq_decls(env);
bool has_heq = has_heq_decls(env);
for (inductive_decl const & d : decls) {
name const & n = inductive_decl_name(d);
pos_info pos = *m_decl_pos_map.find(n);
@ -647,6 +652,14 @@ struct inductive_cmd_fn {
if (has_unit) {
env = mk_cases_on(env, inductive_decl_name(d));
save_def_info(name(n, "cases_on"), pos);
if (has_eq && has_heq) {
env = mk_no_confusion(env, inductive_decl_name(d));
name no_confusion_type_name{n, "no_confusion_type"};
if (env.find(no_confusion_type_name)) {
save_def_info(no_confusion_type_name, pos);
// save_def_info(name(n, "no_confusion"), pos);
}
}
}
}
return env;

4
src/library/definitional/CMakeLists.txt
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@ -1,4 +1,4 @@
add_library(definitional rec_on.cpp induction_on.cpp cases_on.cpp unit.cpp eq.cpp
projection.cpp)
add_library(definitional rec_on.cpp induction_on.cpp cases_on.cpp unit.cpp eq.cpp heq.cpp
no_confusion.cpp projection.cpp)
target_link_libraries(definitional ${LEAN_LIBS})

27
src/library/definitional/heq.cpp Normal file
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@ -0,0 +1,27 @@
/*
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
*/
#include "kernel/environment.h"
namespace lean {
bool has_heq_decls(environment const & env) {
auto d = env.find(name({"heq", "refl"}));
if (!d)
return false;
if (length(d->get_univ_params()) != 1)
return false;
expr type = d->get_type();
for (unsigned i = 0; i < 2; i++) {
if (!is_pi(type))
return type;
type = binding_body(type);
}
type = get_app_fn(type);
if (!is_constant(type))
return false;
return const_name(type) == "heq";
}
}

12
src/library/definitional/heq.h Normal file
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@ -0,0 +1,12 @@
/*
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
*/
#pragma once
#include "kernel/environment.h"
namespace lean {
bool has_heq_decls(environment const & env);
}

144
src/library/definitional/no_confusion.cpp Normal file
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@ -0,0 +1,144 @@
/*
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
*/
#include "util/sstream.h"
#include "kernel/environment.h"
#include "kernel/instantiate.h"
#include "kernel/abstract.h"
#include "kernel/inductive/inductive.h"
#include "kernel/type_checker.h"
#include "library/protected.h"
#include "library/module.h"
namespace lean {
static void throw_corrupted(name const & n) {
throw exception(sstream() << "error in 'no_confusion' generation, '" << n << "' inductive datatype declaration is corrupted");
}
environment mk_no_confusion_type(environment const & env, name const & n) {
optional<inductive::inductive_decls> decls = inductive::is_inductive_decl(env, n);
if (!decls)
throw exception(sstream() << "error in 'no_confusion' generation, '" << n << "' is not an inductive datatype");
name_generator ngen;
unsigned nparams = std::get<1>(*decls);
declaration ind_decl = env.get(n);
declaration cases_decl = env.get(name(n, "cases_on"));
level_param_names lps = cases_decl.get_univ_params();
level rlvl = mk_param_univ(head(lps));
levels ilvls = param_names_to_levels(tail(lps));
if (length(ilvls) != length(ind_decl.get_univ_params()))
return env; // type is a proposition
expr ind_type = instantiate_type_univ_params(ind_decl, ilvls);
name eq_name("eq");
name heq_name("heq");
name eq_refl_name{"eq", "refl"};
name heq_refl_name{"heq", "refl"};
// All inductive datatype parameters and indices are arguments
buffer<expr> args;
while (is_pi(ind_type)) {
expr arg = mk_local(ngen.next(), binding_name(ind_type), binding_domain(ind_type), mk_implicit_binder_info());
args.push_back(arg);
ind_type = instantiate(binding_body(ind_type), arg);
}
if (!is_sort(ind_type) || args.size() < nparams)
throw_corrupted(n);
if (env.impredicative() && is_zero(sort_level(ind_type)))
return env; // do nothing, inductive type is a proposition
unsigned nindices = args.size() - nparams;
// Create inductive datatype
expr I = mk_app(mk_constant(n, ilvls), args);
// Add (P : Type)
expr P = mk_local(ngen.next(), "P", mk_sort(rlvl), binder_info());
args.push_back(P);
// add v1 and v2 elements of the inductive type
expr v1 = mk_local(ngen.next(), "v1", I, binder_info());
expr v2 = mk_local(ngen.next(), "v2", I, binder_info());
args.push_back(v1);
args.push_back(v2);
expr R = mk_sort(rlvl);
name no_confusion_type_name{n, "no_confusion_type"};
expr no_confusion_type_type = Pi(args, R);
// Create type former
buffer<expr> type_former_args;
for (unsigned i = nparams; i < nparams + nindices; i++)
type_former_args.push_back(args[i]);
type_former_args.push_back(v1);
expr type_former = Fun(type_former_args, R);
// Create cases_on
levels clvls = levels(mk_succ(rlvl), ilvls);
expr cases_on = mk_app(mk_app(mk_constant(cases_decl.get_name(), clvls), nparams, args.data()), type_former);
cases_on = mk_app(cases_on, nindices, args.data() + nparams);
expr cases_on1 = mk_app(cases_on, v1);
expr cases_on2 = mk_app(cases_on, v2);
type_checker tc(env);
expr t1 = tc.infer(cases_on1).first;
expr t2 = tc.infer(cases_on2).first;
buffer<expr> outer_cases_on_args;
unsigned idx1 = 0;
while (is_pi(t1)) {
expr minor1 = tc.whnf(binding_domain(t1)).first;
buffer<expr> minor1_args;
while (is_pi(minor1)) {
expr arg = mk_local(ngen.next(), binding_name(minor1), binding_domain(minor1), binding_info(minor1));
minor1_args.push_back(arg);
minor1 = tc.whnf(instantiate(binding_body(minor1), arg)).first;
}
expr curr_t2 = t2;
buffer<expr> inner_cases_on_args;
unsigned idx2 = 0;
while (is_pi(curr_t2)) {
expr minor2 = tc.whnf(binding_domain(curr_t2)).first;
buffer<expr> minor2_args;
while (is_pi(minor2)) {
expr arg = mk_local(ngen.next(), binding_name(minor2), binding_domain(minor2), binding_info(minor2));
minor2_args.push_back(arg);
minor2 = tc.whnf(instantiate(binding_body(minor2), arg)).first;
}
if (idx1 != idx2) {
// infeasible case, constructors do not match
inner_cases_on_args.push_back(Fun(minor2_args, P));
} else {
if (minor1_args.size() != minor2_args.size())
throw_corrupted(n);
buffer<expr> rtype_hyp;
// add equalities
for (unsigned i = 0; i < minor1_args.size(); i++) {
expr lhs = minor1_args[i];
expr rhs = minor2_args[i];
expr lhs_type = mlocal_type(lhs);
expr rhs_type = mlocal_type(rhs);
level l = sort_level(tc.ensure_type(lhs_type).first);
expr h_type;
if (tc.is_def_eq(lhs_type, rhs_type).first) {
h_type = mk_app(mk_constant(eq_name, to_list(l)), lhs_type, lhs, rhs);
} else {
h_type = mk_app(mk_constant(heq_name, to_list(l)), lhs_type, lhs, rhs_type, rhs);
}
rtype_hyp.push_back(mk_local(ngen.next(), local_pp_name(lhs).append_after("_eq"), h_type, binder_info()));
}
inner_cases_on_args.push_back(Fun(minor2_args, mk_arrow(Pi(rtype_hyp, P), P)));
}
idx2++;
curr_t2 = binding_body(curr_t2);
}
outer_cases_on_args.push_back(Fun(minor1_args, mk_app(cases_on2, inner_cases_on_args)));
idx1++;
t1 = binding_body(t1);
}
expr no_confusion_type_value = Fun(args, mk_app(cases_on1, outer_cases_on_args));
bool opaque = false;
bool use_conv_opt = true;
declaration new_d = mk_definition(env, no_confusion_type_name, lps, no_confusion_type_type, no_confusion_type_value,
opaque, ind_decl.get_module_idx(), use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
return add_protected(new_env, no_confusion_type_name);
}
environment mk_no_confusion(environment const & env, name const & n) {
return mk_no_confusion_type(env, n);
}
}

17
src/library/definitional/no_confusion.h Normal file
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@ -0,0 +1,17 @@
/*
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
*/
#pragma once
#include "kernel/environment.h"
namespace lean {
/** \brief Given an inductive datatype \c n (which is not a proposition) in \c env, add
<tt>n.no_confusion_type</tt> and <tt>n.no_confusion</tt> to the environment.
\remark This procedure assumes the environment contains eq, heq, n.cases_on</tt>
*/
environment mk_no_confusion(environment const & env, name const & n);
}

12
tests/lean/no_confusion_type.lean Normal file
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@ -0,0 +1,12 @@
import logic data.nat.basic
open nat
inductive vector (A : Type) : nat → Type :=
vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
check vector.no_confusion_type
constants a1 a2 : num
constants v1 v2 : vector num 2
constant P : Type₁
eval vector.no_confusion_type P (vector.vcons a1 v1) (vector.vcons a2 v2)

2
tests/lean/no_confusion_type.lean.expected.out Normal file
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@ -0,0 +1,2 @@
vector.no_confusion_type : Type → vector ?A ?a → vector ?A ?a → Type
(succ (succ zero) = succ (succ zero) → a1 = a2 → v1 == v2 → P) → P

15
tests/lean/run/nested_begin.lean
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@ -6,16 +6,6 @@ vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
namespace vector
definition no_confusion_type {A : Type} {n : nat} (P : Type) (v₁ v₂ : vector A n) : Type :=
cases_on v₁
(cases_on v₂
(P → P)
(λ n₂ h₂ t₂, P))
(λ n₁ h₁ t₁, cases_on v₂
P
(λ n₂ h₂ t₂, (Π (H : n₁ = n₂), h₁ = h₂ → eq.rec_on H t₁ = t₂ → P) → P))
definition no_confusion {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
assume H₁₂ : v₁ = v₂,
begin
@ -28,7 +18,8 @@ namespace vector
intros (n, a, v, h),
apply (h rfl),
repeat (apply rfl)
repeat (apply rfl),
repeat (apply heq.refl)
end,
eq.rec_on H₁₂ aux H₁₂
end
@ -38,7 +29,7 @@ namespace vector
intro h, apply (no_confusion h), intros, assumption
end
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
begin
intro h, apply (no_confusion h), intros, eassumption
end

9
tests/lean/run/tree.lean
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@ -36,15 +36,6 @@ namespace tree
pr₁ general
end
definition no_confusion_type {A : Type} (P : Type) (t₁ t₂ : tree A) : Type :=
cases_on t₁
(λ a₁, cases_on t₂
(λ a₂, (a₁ = a₂ → P) → P)
(λ l₂ r₂, P))
(λ l₁ r₁, cases_on t₂
(λ a₂, P)
(λ l₂ r₂, (l₁ = l₂ → r₁ = r₂ → P) → P))
set_option pp.universes true
check no_confusion_type

9
tests/lean/run/tree2.lean
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@ -36,15 +36,6 @@ namespace tree
pr₁ general
end
definition no_confusion_type {A : Type} (P : Type) (t₁ t₂ : tree A) : Type :=
cases_on t₁
(λ a₁, cases_on t₂
(λ a₂, (a₁ = a₂ → P) → P)
(λ l₂ r₂, P))
(λ l₁ r₁, cases_on t₂
(λ a₂, P)
(λ l₂ r₂, (l₁ = l₂ → r₁ = r₂ → P) → P))
set_option pp.universes true
check no_confusion_type

9
tests/lean/run/tree3.lean
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@ -36,15 +36,6 @@ namespace tree
pr₁ general
end
definition no_confusion_type {A : Type} (P : Type) (t₁ t₂ : tree A) : Type :=
cases_on t₁
(λ a₁, cases_on t₂
(λ a₂, (a₁ = a₂ → P) → P)
(λ l₂ r₂, P))
(λ l₁ r₁, cases_on t₂
(λ a₂, P)
(λ l₂ r₂, (l₁ = l₂ → r₁ = r₂ → P) → P))
set_option pp.universes true
check no_confusion_type

15
tests/lean/run/vector.lean
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@ -6,22 +6,13 @@ vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
namespace vector
definition no_confusion_type {A : Type} {n : nat} (P : Type) (v₁ v₂ : vector A n) : Type :=
cases_on v₁
(cases_on v₂
(P → P)
(λ n₂ h₂ t₂, P))
(λ n₁ h₁ t₁, cases_on v₂
P
(λ n₂ h₂ t₂, (Π (H : n₁ = n₂), h₁ = h₂ → eq.rec_on H t₁ = t₂ → P) → P))
definition no_confusion {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
assume H₁₂ : v₁ = v₂,
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
take H₁₁, cases_on v₁
(assume h : P, h)
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ = t₁ → P),
h rfl rfl rfl),
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ == t₁ → P),
h rfl rfl !heq.refl),
eq.rec aux H₁₂ H₁₂
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
@ -29,7 +20,7 @@ namespace vector
intro h, apply (no_confusion h), intros, assumption
end
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
begin
intro h, apply (no_confusion h), intros, eassumption
end

13
tests/lean/run/vector2.lean
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@ -6,22 +6,13 @@ vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
namespace vector
definition no_confusion_type {A : Type} {n : nat} (P : Type) (v₁ v₂ : vector A n) : Type :=
cases_on v₁
(cases_on v₂
(P → P)
(λ n₂ h₂ t₂, P))
(λ n₁ h₁ t₁, cases_on v₂
P
(λ n₂ h₂ t₂, (Π (H : n₁ = n₂), h₁ = h₂ → eq.rec_on H t₁ = t₂ → P) → P))
definition no_confusion {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
assume H₁₂ : v₁ = v₂,
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
take H₁₁, cases_on v₁
(assume h : P, h)
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ = t₁ → P),
h rfl rfl rfl),
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ == t₁ → P),
h rfl rfl !heq.refl),
eq.rec aux H₁₂ H₁₂
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=

15
tests/lean/run/vector3.lean
View file

@ -6,28 +6,19 @@ vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
namespace vector
definition no_confusion_type {A : Type} {n : nat} (P : Type) (v₁ v₂ : vector A n) : Type :=
cases_on v₁
(cases_on v₂
(P → P)
(λ n₂ h₂ t₂, P))
(λ n₁ h₁ t₁, cases_on v₂
P
(λ n₂ h₂ t₂, (Π (H : n₁ = n₂), h₁ = h₂ → eq.rec_on H t₁ = t₂ → P) → P))
definition no_confusion {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
assume H₁₂ : v₁ = v₂,
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
take H₁₁, cases_on v₁
(assume h : P, h)
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ = t₁ → P),
h rfl rfl rfl),
(take n₁ h₁ t₁, assume h : (Π (H : n₁ = n₁), h₁ = h₁ → t₁ == t₁ → P),
h rfl rfl !heq.refl),
eq.rec aux H₁₂ H₁₂
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
assume h, no_confusion h (λ n h t, h)
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
assume h, no_confusion h (λ n h t, t)
end vector