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feat(library/logic/if): add dependent if-then-else: dite

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Leonardo de Moura 2014-11-22 09:56:09 -08:00
parent 9c9f5bba1a
commit 47b6cfb28d
2 changed files with 37 additions and 14 deletions
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49
library/logic/if.lean
View file

@ -8,26 +8,26 @@ import logic.decidable tools.tactic
open decidable tactic eq.ops
definition ite (c : Prop) [H : decidable c] {A : Type} (t e : A) : A :=
decidable.rec_on H (assume Hc, t) (assume Hnc, e)
decidable.rec_on H (λ Hc, t) (λ Hnc, e)
notation `if` c `then` t:45 `else` e:45 := ite c t e
definition if_pos {c : Prop} [H : decidable c] (Hc : c) {A : Type} {t e : A} : (if c then t else e) = t :=
decidable.rec
(assume Hc : c, eq.refl (@ite c (inl Hc) A t e))
(assume Hnc : ¬c, absurd Hc Hnc)
(λ Hc : c, eq.refl (@ite c (inl Hc) A t e))
(λ Hnc : ¬c, absurd Hc Hnc)
H
definition if_neg {c : Prop} [H : decidable c] (Hnc : ¬c) {A : Type} {t e : A} : (if c then t else e) = e :=
decidable.rec
(assume Hc : c, absurd Hc Hnc)
(assume Hnc : ¬c, eq.refl (@ite c (inr Hnc) A t e))
(λ Hc : c, absurd Hc Hnc)
(λ Hnc : ¬c, eq.refl (@ite c (inr Hnc) A t e))
H
definition if_t_t (c : Prop) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t :=
decidable.rec
(assume Hc : c, eq.refl (@ite c (inl Hc) A t t))
(assume Hnc : ¬c, eq.refl (@ite c (inr Hnc) A t t))
(λ Hc : c, eq.refl (@ite c (inl Hc) A t t))
(λ Hnc : ¬c, eq.refl (@ite c (inr Hnc) A t t))
H
definition if_true {A : Type} (t e : A) : (if true then t else e) = t :=
@ -39,12 +39,12 @@ if_neg not_false_trivial
theorem if_cond_congr {c₁ c₂ : Prop} [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) {A : Type} (t e : A)
: (if c₁ then t else e) = (if c₂ then t else e) :=
decidable.rec_on H₁
(assume Hc₁ : c₁, decidable.rec_on H₂
(assume Hc₂ : c₂, if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
(assume Hnc₂ : ¬c₂, absurd (iff.elim_left Heq Hc₁) Hnc₂))
(assume Hnc₁ : ¬c₁, decidable.rec_on H₂
(assume Hc₂ : c₂, absurd (iff.elim_right Heq Hc₂) Hnc₁)
(assume Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
(λ Hc₁ : c₁, decidable.rec_on H₂
(λ Hc₂ : c₂, if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
(λ Hnc₂ : ¬c₂, absurd (iff.elim_left Heq Hc₁) Hnc₂))
(λ Hnc₁ : ¬c₁, decidable.rec_on H₂
(λ Hc₂ : c₂, absurd (iff.elim_right Heq Hc₂) Hnc₁)
(λ Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
theorem if_congr_aux {c₁ c₂ : Prop} [H₁ : decidable c₁] [H₂ : decidable c₂] {A : Type} {t₁ t₂ e₁ e₂ : A}
(Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
@ -55,3 +55,26 @@ theorem if_congr {c₁ c₂ : Prop} [H₁ : decidable c₁] {A : Type} {t₁ t
(if c₁ then t₁ else e₁) = (@ite c₂ (decidable_iff_equiv H₁ Hc) A t₂ e₂) :=
have H2 [visible] : decidable c₂, from (decidable_iff_equiv H₁ Hc),
if_congr_aux Hc Ht He
-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
definition dite (c : Prop) [H : decidable c] {A : Type} (t : c → A) (e : ¬ c → A) : A :=
decidable.rec_on H (λ Hc, t Hc) (λ Hnc, e Hnc)
notation `dif` c `then` t:45 `else` e:45 := dite c t e
definition dif_pos {c : Prop} [H : decidable c] (Hc : c) {A : Type} {t : c → A} {e : ¬ c → A} : (dif c then t else e) = t Hc :=
decidable.rec
(λ Hc : c, eq.refl (@dite c (inl Hc) A t e))
(λ Hnc : ¬c, absurd Hc Hnc)
H
definition dif_neg {c : Prop} [H : decidable c] (Hnc : ¬c) {A : Type} {t : c → A} {e : ¬ c → A} : (dif c then t else e) = e Hnc :=
decidable.rec
(λ Hc : c, absurd Hc Hnc)
(λ Hnc : ¬c, eq.refl (@dite c (inr Hnc) A t e))
H
-- Remark: dite and ite are "definitionally equal" when we ignore the proofs.
theorem dite_ite_eq (c : Prop) [H : decidable c] {A : Type} (t : A) (e : A) : dite c (λh, t) (λh, e) = ite c t e :=
rfl

2
src/emacs/lean-syntax.el
View file

@ -96,7 +96,7 @@
(defconst lean-font-lock-defaults
`((;; Keywords
(,(rx word-start (or "calc" "have" "obtains" "show" "by" "in" "let" "forall" "fun"
"exists" "if" "then" "else" "assume" "take" "obtain" "from") word-end)
"exists" "if" "dif" "then" "else" "assume" "take" "obtain" "from") word-end)
. font-lock-keyword-face)
;; String
("\"[^\"]*\"" . 'font-lock-string-face)