michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor(library/data/encodable): mark auxiliary theorems as private

Browse source
This commit is contained in:
Leonardo de Moura 2015-10-16 13:53:08 -07:00
parent f14d0523ec
commit 13ccbaa0d9
1 changed files with 23 additions and 23 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

46
library/data/encodable.lean
View file

@ -50,11 +50,11 @@ variables {A B : Type}
variables [h₁ : encodable A] [h₂ : encodable B] variables [h₁ : encodable A] [h₂ : encodable B]
include h₁ h₂ include h₁ h₂
definition encode_sum : sum A B → nat private definition encode_sum : sum A B → nat
| (sum.inl a) := 2 * encode a | (sum.inl a) := 2 * encode a
| (sum.inr b) := 2 * encode b + 1 | (sum.inr b) := 2 * encode b + 1
definition decode_sum (n : nat) : option (sum A B) := private definition decode_sum (n : nat) : option (sum A B) :=
if n mod 2 = 0 then if n mod 2 = 0 then
match decode A (n div 2) with match decode A (n div 2) with
| some a := some (sum.inl a) | some a := some (sum.inl a)
@ -67,7 +67,7 @@ else
end end
open decidable open decidable
theorem decode_encode_sum : ∀ s : sum A B, decode_sum (encode_sum s) = some s private theorem decode_encode_sum : ∀ s : sum A B, decode_sum (encode_sum s) = some s
| (sum.inl a) := | (sum.inl a) :=
assert aux : 2 > (0:nat), from dec_trivial, assert aux : 2 > (0:nat), from dec_trivial,
begin begin
@ -96,10 +96,10 @@ variables {A B : Type}
variables [h₁ : encodable A] [h₂ : encodable B] variables [h₁ : encodable A] [h₂ : encodable B]
include h₁ h₂ include h₁ h₂
definition encode_prod : A × B → nat private definition encode_prod : A × B → nat
| (a, b) := mkpair (encode a) (encode b) | (a, b) := mkpair (encode a) (encode b)
definition decode_prod (n : nat) : option (A × B) := private definition decode_prod (n : nat) : option (A × B) :=
match unpair n with match unpair n with
| (n₁, n₂) := | (n₁, n₂) :=
match decode A n₁ with match decode A n₁ with
@ -112,7 +112,7 @@ match unpair n with
end end
end end
theorem decode_encode_prod : ∀ p : A × B, decode_prod (encode_prod p) = some p private theorem decode_encode_prod : ∀ p : A × B, decode_prod (encode_prod p) = some p
| (a, b) := | (a, b) :=
begin begin
esimp [encode_prod, decode_prod, prod.cases_on], esimp [encode_prod, decode_prod, prod.cases_on],
@ -133,17 +133,17 @@ variables {A : Type}
variables [h : encodable A] variables [h : encodable A]
include h include h
definition encode_list_core : list A → nat private definition encode_list_core : list A → nat
| [] := 0 | [] := 0
| (a::l) := mkpair (encode a) (encode_list_core l) | (a::l) := mkpair (encode a) (encode_list_core l)
theorem encode_list_core_cons (a : A) (l : list A) : encode_list_core (a::l) = mkpair (encode a) (encode_list_core l) := private theorem encode_list_core_cons (a : A) (l : list A) : encode_list_core (a::l) = mkpair (encode a) (encode_list_core l) :=
rfl rfl
definition encode_list (l : list A) : nat := private definition encode_list (l : list A) : nat :=
mkpair (length l) (encode_list_core l) mkpair (length l) (encode_list_core l)
definition decode_list_core : nat → nat → option (list A) private definition decode_list_core : nat → nat → option (list A)
| 0 v := some [] | 0 v := some []
| (succ n) v := | (succ n) v :=
match unpair v with match unpair v with
@ -158,7 +158,7 @@ definition decode_list_core : nat → nat → option (list A)
end end
end end
theorem decode_list_core_succ (n v : nat) : private theorem decode_list_core_succ (n v : nat) :
decode_list_core (succ n) v = decode_list_core (succ n) v =
match unpair v with match unpair v with
| (v₁, v₂) := | (v₁, v₂) :=
@ -173,12 +173,12 @@ theorem decode_list_core_succ (n v : nat) :
end end
:= rfl := rfl
definition decode_list (n : nat) : option (list A) := private definition decode_list (n : nat) : option (list A) :=
match unpair n with match unpair n with
| (l, v) := decode_list_core l v | (l, v) := decode_list_core l v
end end
theorem decode_encode_list_core : ∀ l : list A, decode_list_core (length l) (encode_list_core l) = some l private theorem decode_encode_list_core : ∀ l : list A, decode_list_core (length l) (encode_list_core l) = some l
| [] := rfl | [] := rfl
| (a::l) := | (a::l) :=
begin begin
@ -189,7 +189,7 @@ theorem decode_encode_list_core : ∀ l : list A, decode_list_core (length l) (e
rewrite [encodable.encodek], rewrite [encodable.encodek],
end end
theorem decode_encode_list (l : list A) : decode_list (encode_list l) = some l := private theorem decode_encode_list (l : list A) : decode_list (encode_list l) = some l :=
begin begin
esimp [encode_list, decode_list], esimp [encode_list, decode_list],
rewrite [unpair_mkpair], rewrite [unpair_mkpair],
@ -240,7 +240,7 @@ open subtype perm
private lemma sorted_eq_of_perm {l₁ l₂ : list A} (h : l₁ ~ l₂) : ensort l₁ = ensort l₂ := private lemma sorted_eq_of_perm {l₁ l₂ : list A} (h : l₁ ~ l₂) : ensort l₁ = ensort l₂ :=
list.sort_eq_of_perm_core enle.total enle.trans enle.refl enle.antisymm h list.sort_eq_of_perm_core enle.total enle.trans enle.refl enle.antisymm h
definition encode_finset (s : finset A) : nat := private definition encode_finset (s : finset A) : nat :=
quot.lift_on s quot.lift_on s
(λ l, encode (ensort (elt_of l))) (λ l, encode (ensort (elt_of l)))
(λ l₁ l₂ p, (λ l₁ l₂ p,
@ -248,13 +248,13 @@ quot.lift_on s
assert ensort (elt_of l₁) = ensort (elt_of l₂), from sorted_eq_of_perm this, assert ensort (elt_of l₁) = ensort (elt_of l₂), from sorted_eq_of_perm this,
by rewrite this) by rewrite this)
definition decode_finset (n : nat) : option (finset A) := private definition decode_finset (n : nat) : option (finset A) :=
match decode (list A) n with match decode (list A) n with
| some l₁ := some (finset.to_finset l₁) | some l₁ := some (finset.to_finset l₁)
| none := none | none := none
end end
theorem decode_encode_finset (s : finset A) : decode_finset (encode_finset s) = some s := private theorem decode_encode_finset (s : finset A) : decode_finset (encode_finset s) = some s :=
quot.induction_on s (λ l, quot.induction_on s (λ l,
begin begin
unfold encode_finset, unfold decode_finset, rewrite encodek, esimp, congruence, unfold encode_finset, unfold decode_finset, rewrite encodek, esimp, congruence,
@ -280,17 +280,17 @@ variable [encA : encodable A]
variable [decP : decidable_pred P] variable [decP : decidable_pred P]
include encA include encA
definition encode_subtype : {a : A | P a} → nat private definition encode_subtype : {a : A | P a} → nat
| (tag v h) := encode v | (tag v h) := encode v
include decP include decP
definition decode_subtype (v : nat) : option {a : A | P a} := private definition decode_subtype (v : nat) : option {a : A | P a} :=
match decode A v with match decode A v with
| some a := if h : P a then some (tag a h) else none | some a := if h : P a then some (tag a h) else none
| none := none | none := none
end end
lemma decode_encode_subtype : ∀ s : {a : A | P a}, decode_subtype (encode_subtype s) = some s private lemma decode_encode_subtype : ∀ s : {a : A | P a}, decode_subtype (encode_subtype s) = some s
| (tag v h) := | (tag v h) :=
begin begin
unfold [encode_subtype, decode_subtype], rewrite encodek, esimp, unfold [encode_subtype, decode_subtype], rewrite encodek, esimp,
@ -449,16 +449,16 @@ choose (exists_rep q)
theorem rep_spec (q : quot s) : ⟦rep q⟧ = q := theorem rep_spec (q : quot s) : ⟦rep q⟧ = q :=
choose_spec (exists_rep q) choose_spec (exists_rep q)
definition encode_quot (q : quot s) : nat := private definition encode_quot (q : quot s) : nat :=
encode (rep q) encode (rep q)
definition decode_quot (n : nat) : option (quot s) := private definition decode_quot (n : nat) : option (quot s) :=
match decode A n with match decode A n with
| some a := some ⟦ a ⟧ | some a := some ⟦ a ⟧
| none := none | none := none
end end
lemma decode_encode_quot (q : quot s) : decode_quot (encode_quot q) = some q := private lemma decode_encode_quot (q : quot s) : decode_quot (encode_quot q) = some q :=
quot.induction_on q (λ l, begin unfold [encode_quot, decode_quot], rewrite encodek, esimp, rewrite rep_spec end) quot.induction_on q (λ l, begin unfold [encode_quot, decode_quot], rewrite encodek, esimp, rewrite rep_spec end)
definition encodable_quot : encodable (quot s) := definition encodable_quot : encodable (quot s) :=