diff --git a/library/logic/if.lean b/library/logic/if.lean index 1750251517..d96f7e49ab 100644 --- a/library/logic/if.lean +++ b/library/logic/if.lean @@ -12,28 +12,28 @@ decidable.rec_on H (assume Hc, t) (assume Hnc, e) notation `if` c `then` t:45 `else` e:45 := ite c t e -theorem if_pos {c : Prop} [H : decidable c] (Hc : c) {A : Type} {t e : A} : (if c then t else e) = t := +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) H -theorem if_neg {c : Prop} [H : decidable c] (Hnc : ¬c) {A : Type} {t e : A} : (if c then t else e) = e := +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)) H -theorem if_t_t (c : Prop) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t := +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)) H -theorem if_true {A : Type} (t e : A) : (if true then t else e) = t := +definition if_true {A : Type} (t e : A) : (if true then t else e) = t := if_pos trivial -theorem if_false {A : Type} (t e : A) : (if false then t else e) = e := +definition if_false {A : Type} (t e : A) : (if false then t else e) = e := 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)