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refactor(library,hott): rename theorems for decidable and inhabited

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The convention is this: we use e.g. nat.is_inhabited and nat.has_decidable_eq
for these two purposes only, to avoid clashing with "inhabited" and "decidable_eq"
in a namespace. Otherwise, we use "decidable_foo" and "inhabited_foo".
This commit is contained in:
Jeremy Avigad 2015-02-25 15:34:49 -05:00 committed by Leonardo de Moura
parent e555531eb6
commit e513b0ead4
8 changed files with 32 additions and 32 deletions
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18
hott/init/logic.hlean
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@ -185,10 +185,10 @@ namespace inhabited
protected definition destruct {A : Type} {B : Type} (H1 : inhabited A) (H2 : A → B) : B :=
inhabited.rec H2 H1
definition fun_inhabited [instance] (A : Type) {B : Type} [H : inhabited B] : inhabited (A → B) :=
definition inhabited_fun [instance] (A : Type) {B : Type} [H : inhabited B] : inhabited (A → B) :=
destruct H (λb, mk (λa, b))
definition dfun_inhabited [instance] (A : Type) {B : A → Type} [H : Πx, inhabited (B x)] :
definition inhabited_Pi [instance] (A : Type) {B : A → Type} [H : Πx, inhabited (B x)] :
inhabited (Πx, B x) :=
mk (λa, destruct (H a) (λb, b))
@ -235,39 +235,39 @@ section
variables {p q : Type}
open decidable (rec_on inl inr)
definition unit.decidable [instance] : decidable unit :=
definition decidable_unit [instance] : decidable unit :=
inl unit.star
definition empty.decidable [instance] : decidable empty :=
definition decidable_empty [instance] : decidable empty :=
inr not_empty
definition prod.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (prod p q) :=
definition decidable_prod [instance] [Hp : decidable p] [Hq : decidable q] : decidable (prod p q) :=
rec_on Hp
(assume Hp : p, rec_on Hq
(assume Hq : q, inl (prod.mk Hp Hq))
(assume Hnq : ¬q, inr (λ H : prod p q, prod.rec_on H (λ Hp Hq, absurd Hq Hnq))))
(assume Hnp : ¬p, inr (λ H : prod p q, prod.rec_on H (λ Hp Hq, absurd Hp Hnp)))
definition sum.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (sum p q) :=
definition decidable_sum [instance] [Hp : decidable p] [Hq : decidable q] : decidable (sum p q) :=
rec_on Hp
(assume Hp : p, inl (sum.inl Hp))
(assume Hnp : ¬p, rec_on Hq
(assume Hq : q, inl (sum.inr Hq))
(assume Hnq : ¬q, inr (λ H : sum p q, sum.rec_on H (λ Hp, absurd Hp Hnp) (λ Hq, absurd Hq Hnq))))
definition not.decidable [instance] [Hp : decidable p] : decidable (¬p) :=
definition decidable_not [instance] [Hp : decidable p] : decidable (¬p) :=
rec_on Hp
(assume Hp, inr (not_not_intro Hp))
(assume Hnp, inl Hnp)
definition implies.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p → q) :=
definition decidable_implies [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p → q) :=
rec_on Hp
(assume Hp : p, rec_on Hq
(assume Hq : q, inl (assume H, Hq))
(assume Hnq : ¬q, inr (assume H : p → q, absurd (H Hp) Hnq)))
(assume Hnp : ¬p, inl (assume Hp, absurd Hp Hnp))
definition iff.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ↔ q) :=
definition decidable_if [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ↔ q) :=
show decidable (prod (p → q) (q → p)), from _
end

4
hott/init/nat.hlean
View file

@ -114,7 +114,7 @@ namespace nat
lt.of_succ_lt hlt),
aux
definition lt.is_decidable_rel [instance] : decidable_rel lt :=
definition decidable_lt [instance] : decidable_rel lt :=
λ a b, nat.rec_on b
(λ (a : nat), inr (not_lt_zero a))
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), nat.cases_on a
@ -145,7 +145,7 @@ namespace nat
(λ hl, eq.rec_on hl !le.refl)
(λ hr, le.of_lt hr)
definition le.is_decidable_rel [instance] : decidable_rel le :=
definition decidable_le [instance] : decidable_rel le :=
λ a b, decidable_iff_equiv _ (iff.intro le.of_eq_or_lt eq_or_lt_of_le)
definition le.rec_on {a : nat} {P : nat → Type} {b : nat} (H : a ≤ b) (H₁ : P a) (H₂ : ∀ b, a < b → P b) : P b :=

4
library/data/bool.lean
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@ -120,10 +120,10 @@ end bool
open bool
protected definition bool.inhabited [instance] : inhabited bool :=
protected definition bool.is_inhabited [instance] : inhabited bool :=
inhabited.mk ff
protected definition bool.decidable_eq [instance] : decidable_eq bool :=
protected definition bool.has_decidable_eq [instance] : decidable_eq bool :=
take a b : bool,
bool.rec_on a
(bool.rec_on b (inl rfl) (inr ff_ne_tt))

2
library/data/list/basic.lean
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@ -215,7 +215,7 @@ list.induction_on l
from H3 ▸ rfl,
!exists.intro (!exists.intro H4)))
definition mem.is_decidable [instance] [H : decidable_eq T] (x : T) (l : list T) : decidable (x ∈ l) :=
definition decidable_mem [instance] [H : decidable_eq T] (x : T) (l : list T) : decidable (x ∈ l) :=
list.rec_on l
(decidable.inr (not_of_iff_false !mem_nil))
(take (h : T) (l : list T) (iH : decidable (x ∈ l)),

10
library/data/sum.lean
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@ -39,15 +39,15 @@ namespace sum
protected definition is_inhabited_right [instance] [h : inhabited B] : inhabited (A + B) :=
inhabited.mk (inr (default B))
protected definition has_eq_decidable [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : ∀ s₁ s₂ : A + B, decidable (s₁ = s₂),
has_eq_decidable (inl a₁) (inl a₂) :=
protected definition has_decidable_eq [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : ∀ s₁ s₂ : A + B, decidable (s₁ = s₂),
has_decidable_eq (inl a₁) (inl a₂) :=
match h₁ a₁ a₂ with
decidable.inl hp := decidable.inl (hp ▸ rfl),
decidable.inr hn := decidable.inr (λ he, absurd (inl_inj he) hn)
end,
has_eq_decidable (inl a₁) (inr b₂) := decidable.inr (λ e, sum.no_confusion e),
has_eq_decidable (inr b₁) (inl a₂) := decidable.inr (λ e, sum.no_confusion e),
has_eq_decidable (inr b₁) (inr b₂) :=
has_decidable_eq (inl a₁) (inr b₂) := decidable.inr (λ e, sum.no_confusion e),
has_decidable_eq (inr b₁) (inl a₂) := decidable.inr (λ e, sum.no_confusion e),
has_decidable_eq (inr b₁) (inr b₂) :=
match h₂ b₁ b₂ with
decidable.inl hp := decidable.inl (hp ▸ rfl),
decidable.inr hn := decidable.inr (λ he, absurd (inr_inj he) hn)

20
library/init/logic.lean
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@ -244,10 +244,10 @@ inductive decidable [class] (p : Prop) : Type :=
inl : p → decidable p,
inr : ¬p → decidable p
definition true.decidable [instance] : decidable true :=
definition decidable_true [instance] : decidable true :=
decidable.inl trivial
definition false.decidable [instance] : decidable false :=
definition decidable_false [instance] : decidable false :=
decidable.inr not_false
namespace decidable
@ -289,33 +289,33 @@ section
variables {p q : Prop}
open decidable (rec_on inl inr)
definition and.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ∧ q) :=
definition decidable_and [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ∧ q) :=
rec_on Hp
(assume Hp : p, rec_on Hq
(assume Hq : q, inl (and.intro Hp Hq))
(assume Hnq : ¬q, inr (assume H : p ∧ q, and.rec_on H (assume Hp Hq, absurd Hq Hnq))))
(assume Hnp : ¬p, inr (assume H : p ∧ q, and.rec_on H (assume Hp Hq, absurd Hp Hnp)))
definition or.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ∨ q) :=
definition decidable_or [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ∨ q) :=
rec_on Hp
(assume Hp : p, inl (or.inl Hp))
(assume Hnp : ¬p, rec_on Hq
(assume Hq : q, inl (or.inr Hq))
(assume Hnq : ¬q, inr (assume H : p ∨ q, or.elim H (assume Hp, absurd Hp Hnp) (assume Hq, absurd Hq Hnq))))
definition not.decidable [instance] [Hp : decidable p] : decidable (¬p) :=
definition decidable_not [instance] [Hp : decidable p] : decidable (¬p) :=
rec_on Hp
(assume Hp, inr (λ Hnp, absurd Hp Hnp))
(assume Hnp, inl Hnp)
definition implies.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p → q) :=
definition decidable_implies [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p → q) :=
rec_on Hp
(assume Hp : p, rec_on Hq
(assume Hq : q, inl (assume H, Hq))
(assume Hnq : ¬q, inr (assume H : p → q, absurd (H Hp) Hnq)))
(assume Hnp : ¬p, inl (assume Hp, absurd Hp Hnp))
definition iff.decidable [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ↔ q) :=
definition decidable_iff [instance] [Hp : decidable p] [Hq : decidable q] : decidable (p ↔ q) :=
show decidable ((p → q) ∧ (q → p)), from _
end
@ -341,13 +341,13 @@ inhabited.rec (λa, a) H
opaque definition arbitrary (A : Type) [H : inhabited A] : A :=
inhabited.rec (λa, a) H
definition Prop_inhabited [instance] : inhabited Prop :=
definition Prop.is_inhabited [instance] : inhabited Prop :=
inhabited.mk true
definition fun_inhabited [instance] (A : Type) {B : Type} [H : inhabited B] : inhabited (A → B) :=
definition inhabited_fun [instance] (A : Type) {B : Type} [H : inhabited B] : inhabited (A → B) :=
inhabited.rec_on H (λb, inhabited.mk (λa, b))
definition dfun_inhabited [instance] (A : Type) {B : A → Type} [H : Πx, inhabited (B x)] :
definition inhabited_Pi [instance] (A : Type) {B : A → Type} [H : Πx, inhabited (B x)] :
inhabited (Πx, B x) :=
inhabited.mk (λa, inhabited.rec_on (H a) (λb, b))

4
library/init/nat.lean
View file

@ -105,7 +105,7 @@ namespace nat
lt_of_succ_lt hlt),
aux
definition lt.is_decidable_rel [instance] : decidable_rel lt :=
definition decidable_lt [instance] : decidable_rel lt :=
λ a b, nat.rec_on b
(λ (a : nat), inr (not_lt_zero a))
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), nat.cases_on a
@ -136,7 +136,7 @@ namespace nat
(λ hl, eq.rec_on hl !le.refl)
(λ hr, le_of_lt hr)
definition le.is_decidable_rel [instance] : decidable_rel le :=
definition decidable_le [instance] : decidable_rel le :=
λ a b, decidable_of_decidable_of_iff _ (iff.intro le_of_eq_or_lt eq_or_lt_of_le)
definition le.rec_on {a : nat} {P : nat → Prop} {b : nat} (H : a ≤ b) (H₁ : P a) (H₂ : ∀ b, a < b → P b) : P b :=

2
library/logic/identities.lean
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@ -106,7 +106,7 @@ theorem exists_not_of_not_forall {A : Type} {P : A → Prop} [D : ∀x, decidabl
[D' : decidable (∃x, ¬P x)] (H : ¬∀x, P x) :
∃x, ¬P x :=
@by_contradiction _ D' (assume H1 : ¬∃x, ¬P x,
have H2 : ∀x, ¬¬P x, from @forall_not_of_not_exists _ _ (take x, not.decidable) H1,
have H2 : ∀x, ¬¬P x, from @forall_not_of_not_exists _ _ (take x, decidable_not) H1,
have H3 : ∀x, P x, from take x, @not_not_elim _ (D x) (H2 x),
absurd H3 H)