feat(library/logic/classes/decidable): generalize 'by_cases' theorem

library/data/nat/basic.lean

@@ -11,7 +11,7 @@ import logic data.num tools.tactic struc.binary tools.helper_tactics
 import logic.classes.inhabited
 
 open tactic binary eq_ops
-open decidable (hiding induction_on rec_on)
+open decidable
 open relation -- for subst_iff
 open helper_tactics
 
@@ -117,8 +117,7 @@ have general : ∀n, decidable (n = m), from
       take n, rec_on n
         (inr (ne.symm succ_ne_zero))
         (λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
-          have d1 : decidable (n' = m'), from iH1 n',
-          decidable.rec_on d1
+          decidable.by_cases
             (assume Heq : n' = m', inl (congr_arg succ Heq))
             (assume Hne : n' ≠ m',
               have H1 : succ n' ≠ succ m', from

library/data/nat/order.lean

@@ -241,8 +241,7 @@ have general : ∀n, decidable (n ≤ m), from
       rec_on n
         (decidable.inl zero_le)
         (take (n' : ℕ) (iH2 : decidable (n' ≤ succ m')),
-          have d1 : decidable (n' ≤ m'), from iH1 n',
-          decidable.rec_on d1
+          decidable.by_cases
             (assume Hp : n' ≤ m', decidable.inl (succ_le Hp))
             (assume Hn : ¬ n' ≤ m',
               have H : ¬ succ n' ≤ succ m', from

library/logic/classes/decidable.lean

@@ -38,8 +38,8 @@ decidable.rec
 theorem em (p : Prop) {H : decidable p} : p ∨ ¬p :=
 induction_on H (λ Hp, or.inl Hp) (λ Hnp, or.inr Hnp)
 
-theorem by_cases {a b : Prop} {C : decidable a} (Hab : a → b) (Hnab : ¬a → b) : b :=
-or.elim (em a) (assume Ha, Hab Ha) (assume Hna, Hnab Hna)
+theorem by_cases {a : Prop} {b : Type} {C : decidable a} (Hab : a → b) (Hnab : ¬a → b) : b :=
+rec_on C (assume Ha, Hab Ha) (assume Hna, Hnab Hna)
 
 theorem by_contradiction {p : Prop} {Hp : decidable p} (H : ¬p → false) : p :=
 or.elim (em p)