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feat(builtin): add sum types

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2014-02-05 23:04:44 -08:00
parent 419fb7464e
commit c01f82aeb7
8 changed files with 133 additions and 10 deletions
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1
src/builtin/CMakeLists.txt
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@ -95,6 +95,7 @@ add_theory("Real.lean" "${CMAKE_CURRENT_BINARY_DIR}/Int.olean")
add_theory("specialfn.lean" "${CMAKE_CURRENT_BINARY_DIR}/Real.olean")
add_theory("subtype.lean" "${CMAKE_CURRENT_BINARY_DIR}/kernel.olean")
add_theory("optional.lean" "${CMAKE_CURRENT_BINARY_DIR}/subtype.olean")
add_theory("sum.lean" "${CMAKE_CURRENT_BINARY_DIR}/optional.olean")
update_interface("kernel.olean" "kernel" "-n")
update_interface("Nat.olean" "library/arith" "-n")

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src/builtin/obj/kernel.olean
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src/builtin/obj/optional.olean
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src/builtin/obj/subtype.olean
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src/builtin/obj/sum.olean Normal file
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16
src/builtin/optional.lean
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@ -20,25 +20,25 @@ theorem some_pred {A : (Type U)} (a : A) : optional_pred A (λ x, x = a)
theorem none_pred (A : (Type U)) : optional_pred A (λ x, false)
:= or_introl (take x, not_false_trivial) (exists_unique (λ x, false))
theorem optional_inhabited (A : (Type U)) : inhabited (optional A)
theorem inhab (A : (Type U)) : inhabited (optional A)
:= subtype_inhabited (exists_intro (λ x, false) (none_pred A))
definition none {A : (Type U)} : optional A
:= abst (λ x, false) (optional_inhabited A)
:= abst (λ x, false) (inhab A)
definition some {A : (Type U)} (a : A) : optional A
:= abst (λ x, x = a) (optional_inhabited A)
:= abst (λ x, x = a) (inhab A)
definition is_some {A : (Type U)} (n : optional A) := ∃ x : A, some x = n
theorem injectivity {A : (Type U)} {a a' : A} : some a = some a' → a = a'
:= assume Heq,
let eq_reps : (λ x, x = a) = (λ x, x = a') := abst_inj (optional_inhabited A) (some_pred a) (some_pred a') Heq
let eq_reps : (λ x, x = a) = (λ x, x = a') := abst_inj (inhab A) (some_pred a) (some_pred a') Heq
in (congr1 a eq_reps) ◂ (refl a)
theorem distinct {A : (Type U)} (a : A) : some a ≠ none
:= assume N : some a = none,
let eq_reps : (λ x, x = a) = (λ x, false) := abst_inj (optional_inhabited A) (some_pred a) (none_pred A) N
let eq_reps : (λ x, x = a) = (λ x, false) := abst_inj (inhab A) (some_pred a) (none_pred A) N
in (congr1 a eq_reps) ◂ (refl a)
definition value {A : (Type U)} (n : optional A) (H : is_some n) : A
@ -69,13 +69,13 @@ theorem dichotomy {A : (Type U)} (n : optional A) : n = none ∨ ∃ a, n = some
:= let pred : optional_pred A (rep n) := P_rep n
in or_elim pred
(λ Hl, let rep_n_eq : rep n = λ x, false := false_pred Hl,
rep_none_eq : rep none = λ x, false := rep_abst (optional_inhabited A) (λ x, false) (none_pred A)
rep_none_eq : rep none = λ x, false := rep_abst (inhab A) (λ x, false) (none_pred A)
in or_introl (rep_inj (trans rep_n_eq (symm rep_none_eq)))
(∃ a, n = some a))
(λ Hr : ∃ x, rep n x ∧ ∀ y, y ≠ x → ¬ rep n y,
obtain (w : A) (Hw : rep n w ∧ ∀ y, y ≠ w → ¬ rep n y), from Hr,
let rep_n_eq : rep n = λ x, x = w := singleton_pred Hw,
rep_some_eq : rep (some w) = λ x, x = w := rep_abst (optional_inhabited A) (λ x, x = w) (some_pred w),
rep_some_eq : rep (some w) = λ x, x = w := rep_abst (inhab A) (λ x, x = w) (some_pred w),
n_eq_some : n = some w := rep_inj (trans rep_n_eq (symm rep_some_eq))
in or_intror (n = none)
(exists_intro w n_eq_some))
@ -96,5 +96,5 @@ end
set_opaque optional true
set_opaque optional_pred true
definition optional_inhabited := optional::optional_inhabited
definition optional_inhabited := optional::inhab
add_rewrite optional::is_some_some optional::not_is_some_none optional::distinct optional::value_some

122
src/builtin/sum.lean Normal file
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@ -0,0 +1,122 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import macros
import subtype
import optional
using subtype
using optional
-- We are encoding the (sum A B) as a subtype of (optional A) ⨯ (optional B), where
-- (proj1 n = none) ≠ (proj2 n = none)
definition sum_pred (A B : (Type U)) := λ p : (optional A) ⨯ (optional B), (proj1 p = none) ≠ (proj2 p = none)
definition sum (A B : (Type U)) := subtype ((optional A) ⨯ (optional B)) (sum_pred A B)
namespace sum
-- TODO: move pair, pair_inj1 and pair_inj2 to separate file
definition pair {A : (Type U)} {B : (Type U)} (a : A) (b : B) := tuple a, b
theorem pair_inj1 {A : (Type U)} {B : A → (Type U)} {a b : sig x, B x} (H : a = b) : proj1 a = proj1 b
:= subst (refl (proj1 a)) H
theorem pair_inj2 {A B : (Type U)} {a b : A ⨯ B} (H : a = b) : proj2 a = proj2 b
:= subst (refl (proj2 a)) H
theorem pairext_proj {A B : (Type U)} {p : A ⨯ B} {a : A} {b : B} (H1 : proj1 p = a) (H2 : proj2 p = b) : p = (pair a b)
:= @pairext A (λ x, B) p (pair a b) H1 (to_heq H2)
theorem inl_pred {A : (Type U)} (a : A) (B : (Type U)) : sum_pred A B (pair (some a) none)
:= not_intro
(λ N : (some a = none) = (none = (optional::@none B)),
let eq : some a = none := (symm N) ◂ (refl (optional::@none B))
in absurd eq (distinct a))
theorem inr_pred (A : (Type U)) {B : (Type U)} (b : B) : sum_pred A B (pair none (some b))
:= not_intro
(λ N : (none = (optional::@none A)) = (some b = none),
let eq : some b = none := N ◂ (refl (optional::@none A))
in absurd eq (distinct b))
theorem inhabl {A : (Type U)} (H : inhabited A) (B : (Type U)) : inhabited (sum A B)
:= inhabited_elim H (λ w : A,
subtype_inhabited (exists_intro (pair (some w) none) (inl_pred w B)))
theorem inhabr (A : (Type U)) {B : (Type U)} (H : inhabited B) : inhabited (sum A B)
:= inhabited_elim H (λ w : B,
subtype_inhabited (exists_intro (pair none (some w)) (inr_pred A w)))
definition inl {A : (Type U)} (a : A) (B : (Type U)) : sum A B
:= abst (pair (some a) none) (inhabl (inhabited_intro a) B)
definition inr (A : (Type U)) {B : (Type U)} (b : B) : sum A B
:= abst (pair none (some b)) (inhabr A (inhabited_intro b))
theorem inl_inj {A : (Type U)} (a1 a2 : A) (B : (Type U)) : inl a1 B = inl a2 B → a1 = a2
:= assume Heq : inl a1 B = inl a2 B,
let eq1 : inl a1 B = abst (pair (some a1) none) (inhabl (inhabited_intro a1) B) := refl (inl a1 B),
eq2 : inl a2 B = abst (pair (some a2) none) (inhabl (inhabited_intro a1) B)
:= subst (refl (inl a2 B)) (proof_irrel (inhabl (inhabited_intro a2) B) (inhabl (inhabited_intro a1) B)),
rep_eq : (pair (some a1) none) = (pair (some a2) none)
:= abst_inj (inhabl (inhabited_intro a1) B) (inl_pred a1 B) (inl_pred a2 B) (trans (trans (symm eq1) Heq) eq2)
in optional::injectivity
(calc some a1 = proj1 (pair (some a1) (optional::@none B)) : refl (some a1)
... = proj1 (pair (some a2) (optional::@none B)) : pair_inj1 rep_eq
... = some a2 : refl (some a2))
theorem inr_inj (A : (Type U)) {B : (Type U)} (b1 b2 : B) : inr A b1 = inr A b2 → b1 = b2
:= assume Heq : inr A b1 = inr A b2,
let eq1 : inr A b1 = abst (pair none (some b1)) (inhabr A (inhabited_intro b1)) := refl (inr A b1),
eq2 : inr A b2 = abst (pair none (some b2)) (inhabr A (inhabited_intro b1))
:= subst (refl (inr A b2)) (proof_irrel (inhabr A (inhabited_intro b2)) (inhabr A (inhabited_intro b1))),
rep_eq : (pair none (some b1)) = (pair none (some b2))
:= abst_inj (inhabr A (inhabited_intro b1)) (inr_pred A b1) (inr_pred A b2) (trans (trans (symm eq1) Heq) eq2)
in optional::injectivity
(calc some b1 = proj2 (pair (optional::@none A) (some b1)) : refl (some b1)
... = proj2 (pair (optional::@none A) (some b2)) : pair_inj2 rep_eq
... = some b2 : refl (some b2))
theorem distinct {A B : (Type U)} (a : A) (b : B) : inl a B ≠ inr A b
:= assume N : inl a B = inr A b,
let eq1 : inl a B = abst (pair (some a) none) (inhabl (inhabited_intro a) B) := refl (inl a B),
eq2 : inr A b = abst (pair none (some b)) (inhabl (inhabited_intro a) B)
:= subst (refl (inr A b)) (proof_irrel (inhabr A (inhabited_intro b)) (inhabl (inhabited_intro a) B)),
rep_eq : (pair (some a) none) = (pair none (some b))
:= abst_inj (inhabl (inhabited_intro a) B) (inl_pred a B) (inr_pred A b) (trans (trans (symm eq1) N) eq2)
in absurd (pair_inj1 rep_eq) (optional::distinct a)
theorem dichotomy {A B : (Type U)} (n : sum A B) : (∃ a, n = inl a B) ∨ (∃ b, n = inr A b)
:= let pred : (proj1 (rep n) = none) ≠ (proj2 (rep n) = none) := P_rep n
in or_elim (em (proj1 (rep n) = none))
(λ Heq, let neq_none : ¬ proj2 (rep n) = (optional::@none B) := (symm (not_iff_elim (ne_symm pred))) ◂ Heq,
ex_some : ∃ b, proj2 (rep n) = some b := resolve1 (optional::dichotomy (proj2 (rep n))) neq_none
in obtain (b : B) (Hb : proj2 (rep n) = some b), from ex_some,
or_intror (∃ a, n = inl a B)
(let rep_eq : rep n = pair none (some b)
:= pairext_proj Heq Hb,
rep_inr : rep (inr A b) = pair none (some b)
:= rep_abst (inhabr A (inhabited_intro b)) (pair none (some b)) (inr_pred A b),
n_eq_inr : n = inr A b
:= rep_inj (trans rep_eq (symm rep_inr))
in exists_intro b n_eq_inr))
(λ Hne, let eq_none : proj2 (rep n) = (optional::@none B) := (not_iff_elim pred) ◂ Hne,
ex_some : ∃ a, proj1 (rep n) = some a := resolve1 (optional::dichotomy (proj1 (rep n))) Hne
in obtain (a : A) (Ha : proj1 (rep n) = some a), from ex_some,
or_introl (let rep_eq : rep n = pair (some a) none
:= pairext_proj Ha eq_none,
rep_inl : rep (inl a B) = pair (some a) none
:= rep_abst (inhabl (inhabited_intro a) B) (pair (some a) none) (inl_pred a B),
n_eq_inl : n = inl a B
:= rep_inj (trans rep_eq (symm rep_inl))
in exists_intro a n_eq_inl)
(∃ b, n = inr A b))
theorem induction {A B : (Type U)} {P : sum A B → Bool} (H1 : ∀ a, P (inl a B)) (H2 : ∀ b, P (inr A b)) : ∀ n, P n
:= take n, or_elim (sum::dichotomy n)
(λ Hex : ∃ a, n = inl a B,
obtain (a : A) (Ha : n = inl a B), from Hex,
subst (H1 a) (symm Ha))
(λ Hex : ∃ b, n = inr A b,
obtain (b : B) (Hb : n = inr A b), from Hex,
subst (H2 b) (symm Hb))
set_opaque inl true
set_opaque inr true
end
set_opaque sum_pred true
set_opaque sum true

4
src/library/elaborator/elaborator.cpp
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@ -1522,8 +1522,6 @@ class elaborator::imp {
}
}
if (!is_meta_app(a) && !is_meta_app(b) && normalize_head(a, b, c)) { return true; }
if (!eq) {
// Try to assign convertability constraints.
if (is_metavar(a) && !is_assigned(a) && !has_local_context(a)) {
@ -1549,6 +1547,8 @@ class elaborator::imp {
}
}
if (!is_meta_app(a) && !is_meta_app(b) && normalize_head(a, b, c)) { return true; }
if (process_simple_ho_match(ctx, a, b, true, c) ||
process_simple_ho_match(ctx, b, a, false, c))
return true;