feat(library): add '[rewrite]' annotation some some theorems

library/data/equiv.lean

@@ -50,22 +50,22 @@ lemma arrow_congr {A₁ B₁ A₂ B₂ : Type} : A₁ ≃ A₂ → B₁ ≃ B₂
 
 section
 open unit
-lemma arrow_unit_equiv_unit (A : Type) : (A → unit) ≃ unit :=
+lemma arrow_unit_equiv_unit [rewrite] (A : Type) : (A → unit) ≃ unit :=
 mk (λ f, star) (λ u, (λ f, star))
    (λ f, funext (λ x, by cases (f x); reflexivity))
    (λ u, by cases u; reflexivity)
 
-lemma unit_arrow_equiv (A : Type) : (unit → A) ≃ A :=
+lemma unit_arrow_equiv [rewrite] (A : Type) : (unit → A) ≃ A :=
 mk (λ f, f star) (λ a, (λ u, a))
    (λ f, funext (λ x, by cases x; reflexivity))
    (λ u, rfl)
 
-lemma empty_arrow_equiv_unit (A : Type) : (empty → A) ≃ unit :=
+lemma empty_arrow_equiv_unit [rewrite] (A : Type) : (empty → A) ≃ unit :=
 mk (λ f, star) (λ u, λ e, empty.rec _ e)
    (λ f, funext (λ x, empty.rec _ x))
    (λ u, by cases u; reflexivity)
 
-lemma false_arrow_equiv_unit (A : Type) : (false → A) ≃ unit :=
+lemma false_arrow_equiv_unit [rewrite] (A : Type) : (false → A) ≃ unit :=
 calc (false → A) ≃ (empty → A) : arrow_congr false_equiv_empty !equiv.refl
              ... ≃ unit        : empty_arrow_equiv_unit
 end
@@ -78,13 +78,13 @@ lemma prod_congr {A₁ B₁ A₂ B₂ : Type} : A₁ ≃ A₂ → B₁ ≃ B₂
     (λ p, begin cases p, esimp, rewrite [l₁, l₂] end)
     (λ p, begin cases p, esimp, rewrite [r₁, r₂] end)
 
-lemma prod_comm (A B : Type) : (A × B) ≃ (B × A) :=
+lemma prod_comm [rewrite] (A B : Type) : (A × B) ≃ (B × A) :=
 mk (λ p, match p with (a, b) := (b, a) end)
    (λ p, match p with (b, a) := (a, b) end)
    (λ p, begin cases p, esimp end)
    (λ p, begin cases p, esimp end)
 
-lemma prod_assoc (A B C : Type) : ((A × B) × C) ≃ (A × (B × C)) :=
+lemma prod_assoc [rewrite] (A B C : Type) : ((A × B) × C) ≃ (A × (B × C)) :=
 mk (λ t, match t with ((a, b), c) := (a, (b, c)) end)
    (λ t, match t with (a, (b, c)) := ((a, b), c) end)
    (λ t, begin cases t with ab c, cases ab, esimp end)
@@ -92,20 +92,20 @@ mk (λ t, match t with ((a, b), c) := (a, (b, c)) end)
 
 section
 open unit prod.ops
-lemma prod_unit_right (A : Type) : (A × unit) ≃ A :=
+lemma prod_unit_right [rewrite] (A : Type) : (A × unit) ≃ A :=
 mk (λ p, p.1)
    (λ a, (a, star))
    (λ p, begin cases p with a u, cases u, esimp end)
    (λ a, rfl)
 
-lemma prod_unit_left (A : Type) : (unit × A) ≃ A :=
+lemma prod_unit_left [rewrite] (A : Type) : (unit × A) ≃ A :=
 calc (unit × A) ≃ (A × unit) : prod_comm
             ... ≃ A          : prod_unit_right
 
-lemma prod_empty_right (A : Type) : (A × empty) ≃ empty :=
+lemma prod_empty_right [rewrite] (A : Type) : (A × empty) ≃ empty :=
 mk (λ p, empty.rec _ p.2) (λ e, empty.rec _ e) (λ p, empty.rec _ p.2)  (λ e, empty.rec _ e)
 
-lemma prod_empty_left (A : Type) : (empty × A) ≃ empty :=
+lemma prod_empty_left [rewrite] (A : Type) : (empty × A) ≃ empty :=
 calc (empty × A) ≃ (A × empty) : prod_comm
              ... ≃ empty       : prod_empty_right
 end
@@ -127,25 +127,25 @@ mk (λ b, match b with tt := inl star | ff := inr star end)
    (λ b, begin cases b, esimp, esimp end)
    (λ s, begin cases s with u u, {cases u, esimp}, {cases u, esimp} end)
 
-lemma sum_comm (A B : Type) : (A + B) ≃ (B + A) :=
+lemma sum_comm [rewrite] (A B : Type) : (A + B) ≃ (B + A) :=
 mk (λ s, match s with inl a := inr a | inr b := inl b end)
    (λ s, match s with inl b := inr b | inr a := inl a end)
    (λ s, begin cases s, esimp, esimp end)
    (λ s, begin cases s, esimp, esimp end)
 
-lemma sum_assoc (A B C : Type) : ((A + B) + C) ≃ (A + (B + C)) :=
+lemma sum_assoc [rewrite] (A B C : Type) : ((A + B) + C) ≃ (A + (B + C)) :=
 mk (λ s, match s with inl (inl a) := inl a | inl (inr b) := inr (inl b) | inr c := inr (inr c) end)
    (λ s, match s with inl a := inl (inl a) | inr (inl b) := inl (inr b) | inr (inr c) := inr c end)
    (λ s, begin cases s with ab c, cases ab, repeat esimp end)
    (λ s, begin cases s with a bc, esimp, cases bc, repeat esimp end)
 
-lemma sum_empty_right (A : Type) : (A + empty) ≃ A :=
+lemma sum_empty_right [rewrite] (A : Type) : (A + empty) ≃ A :=
 mk (λ s, match s with inl a := a | inr e := empty.rec _ e end)
    (λ a, inl a)
    (λ s, begin cases s with a e, esimp, exact empty.rec _ e end)
    (λ a, rfl)
 
-lemma sum_empty_left (A : Type) : (empty + A) ≃ A :=
+lemma sum_empty_left [rewrite] (A : Type) : (empty + A) ≃ A :=
 calc (empty + A) ≃ (A + empty) : sum_comm
           ...    ≃ A           : sum_empty_right
 end
@@ -191,23 +191,23 @@ mk (λ n, match n with 0 := inr star | succ a := inl a end)
    (λ n, begin cases n, repeat esimp end)
    (λ s, begin cases s with a u, esimp, {cases u, esimp} end)
 
-lemma nat_sum_unit_equiv_nat : (nat + unit) ≃ nat :=
+lemma nat_sum_unit_equiv_nat [rewrite] : (nat + unit) ≃ nat :=
 equiv.symm nat_equiv_nat_sum_unit
 
-lemma nat_prod_nat_equiv_nat : (nat × nat) ≃ nat :=
+lemma nat_prod_nat_equiv_nat [rewrite] : (nat × nat) ≃ nat :=
 mk (λ p, mkpair p.1 p.2)
    (λ n, unpair n)
    (λ p, begin cases p, apply unpair_mkpair end)
    (λ n, mkpair_unpair n)
 
-lemma nat_sum_bool_equiv_nat : (nat + bool) ≃ nat :=
+lemma nat_sum_bool_equiv_nat [rewrite] : (nat + bool) ≃ nat :=
 calc (nat + bool) ≃ (nat + (unit + unit)) : sum_congr !equiv.refl bool_equiv_unit_sum_unit
              ...  ≃ ((nat + unit) + unit) : sum_assoc
              ...  ≃ (nat + unit)          : sum_congr nat_sum_unit_equiv_nat !equiv.refl
              ...  ≃ nat                   : nat_sum_unit_equiv_nat
 
 open decidable
-lemma nat_sum_nat_equiv_nat : (nat + nat) ≃ nat :=
+lemma nat_sum_nat_equiv_nat [rewrite] : (nat + nat) ≃ nat :=
 mk (λ s, match s with inl n := 2*n | inr n := 2*n+1 end)
    (λ n, if even n then inl (n div 2) else inr ((n - 1) div 2))
    (λ s, begin

library/data/finset/basic.lean

@@ -84,7 +84,7 @@ theorem mem_list_of_mem {a : A} {l : nodup_list A} : a ∈ ⟦l⟧ → a ∈ elt
 definition singleton (a : A) : finset A :=
 to_finset_of_nodup [a] !nodup_singleton
 
-theorem mem_singleton (a : A) : a ∈ singleton a :=
+theorem mem_singleton [rewrite] (a : A) : a ∈ singleton a :=
 mem_of_mem_list !mem_cons
 
 theorem eq_of_mem_singleton {x a : A} : x ∈ singleton a → x = a :=
@@ -116,10 +116,10 @@ to_finset_of_nodup [] nodup_nil
 
 notation `∅` := !empty
 
-theorem not_mem_empty (a : A) : a ∉ ∅ :=
+theorem not_mem_empty [rewrite] (a : A) : a ∉ ∅ :=
 λ aine : a ∈ ∅, aine
 
-theorem mem_empty_iff (x : A) : x ∈ ∅ ↔ false :=
+theorem mem_empty_iff [rewrite] (x : A) : x ∈ ∅ ↔ false :=
 iff.mp' !iff_false_iff_not !not_mem_empty
 
 theorem mem_empty_eq (x : A) : x ∈ ∅ = false :=

library/data/list/basic.lean

@@ -18,7 +18,7 @@ notation `[` l:(foldr `,` (h t, cons h t) nil `]`) := l
 
 variable {T : Type}
 
-lemma cons_ne_nil (a : T) (l : list T) : a::l ≠ [] :=
+lemma cons_ne_nil [rewrite] (a : T) (l : list T) : a::l ≠ [] :=
 by contradiction
 
 lemma head_eq_of_cons_eq {A : Type} {h₁ h₂ : A} {t₁ t₂ : list A} :
@@ -40,17 +40,17 @@ definition append : list T → list T → list T
 
 notation l₁ ++ l₂ := append l₁ l₂
 
-theorem append_nil_left (t : list T) : [] ++ t = t
+theorem append_nil_left [rewrite] (t : list T) : [] ++ t = t
 
-theorem append_cons (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)
+theorem append_cons [rewrite] (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)
 
-theorem append_nil_right : ∀ (t : list T), t ++ [] = t
+theorem append_nil_right [rewrite] : ∀ (t : list T), t ++ [] = t
 | []       := rfl
 | (a :: l) := calc
   (a :: l) ++ [] = a :: (l ++ []) : rfl
              ... = a :: l         : append_nil_right l
 
-theorem append.assoc : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u)
+theorem append.assoc [rewrite] : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u)
 | []       t u := rfl
 | (a :: l) t u :=
   show a :: (l ++ t ++ u) = (a :: l) ++ (t ++ u),
@@ -61,11 +61,11 @@ definition length : list T → nat
 | []       := 0
 | (a :: l) := length l + 1
 
-theorem length_nil : length (@nil T) = 0
+theorem length_nil [rewrite] : length (@nil T) = 0
 
-theorem length_cons (x : T) (t : list T) : length (x::t) = length t + 1
+theorem length_cons [rewrite] (x : T) (t : list T) : length (x::t) = length t + 1
 
-theorem length_append : ∀ (s t : list T), length (s ++ t) = length s + length t
+theorem length_append [rewrite] : ∀ (s t : list T), length (s ++ t) = length s + length t
 | []       t := calc
     length ([] ++ t)  = length t : rfl
                    ... = length [] + length t : zero_add
@@ -91,9 +91,9 @@ definition concat : Π (x : T), list T → list T
 | a []       := [a]
 | a (b :: l) := b :: concat a l
 
-theorem concat_nil (x : T) : concat x [] = [x]
+theorem concat_nil [rewrite] (x : T) : concat x [] = [x]
 
-theorem concat_cons (x y : T) (l : list T) : concat x (y::l)  = y::(concat x l)
+theorem concat_cons [rewrite] (x y : T) (l : list T) : concat x (y::l)  = y::(concat x l)
 
 theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a]
 | []       := rfl
@@ -101,7 +101,7 @@ theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a]
   show b :: (concat a l) = (b :: l) ++ (a :: []),
   by rewrite concat_eq_append
 
-theorem concat_ne_nil (a : T) : ∀ (l : list T), concat a l ≠ [] :=
+theorem concat_ne_nil [rewrite] (a : T) : ∀ (l : list T), concat a l ≠ [] :=
 by intro l; induction l; repeat contradiction
 
 /- last -/
@@ -111,16 +111,16 @@ definition last : Π l : list T, l ≠ [] → T
 | [a]         h := a
 | (a₁::a₂::l) h := last (a₂::l) !cons_ne_nil
 
-lemma last_singleton (a : T) (h : [a] ≠ []) : last [a] h = a :=
+lemma last_singleton [rewrite] (a : T) (h : [a] ≠ []) : last [a] h = a :=
 rfl
 
-lemma last_cons_cons (a₁ a₂ : T) (l : list T) (h : a₁::a₂::l ≠ []) : last (a₁::a₂::l) h = last (a₂::l) !cons_ne_nil :=
+lemma last_cons_cons [rewrite] (a₁ a₂ : T) (l : list T) (h : a₁::a₂::l ≠ []) : last (a₁::a₂::l) h = last (a₂::l) !cons_ne_nil :=
 rfl
 
 theorem last_congr {l₁ l₂ : list T} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : last l₁ h₁ = last l₂ h₂ :=
 by subst l₁
 
-theorem last_concat {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
+theorem last_concat [rewrite] {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
 | []          h := rfl
 | [a]         h := rfl
 | (a₁::a₂::l) h :=
@@ -139,13 +139,13 @@ definition reverse : list T → list T
 | []       := []
 | (a :: l) := concat a (reverse l)
 
-theorem reverse_nil : reverse (@nil T) = []
+theorem reverse_nil [rewrite] : reverse (@nil T) = []
 
-theorem reverse_cons (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
+theorem reverse_cons [rewrite] (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
 
-theorem reverse_singleton (x : T) : reverse [x] = [x]
+theorem reverse_singleton [rewrite] (x : T) : reverse [x] = [x]
 
-theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s)
+theorem reverse_append [rewrite] : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s)
 | []         t2 := calc
     reverse ([] ++ t2) = reverse t2                    : rfl
                 ...    = (reverse t2) ++ []           : append_nil_right
@@ -158,7 +158,7 @@ theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (
                         ...    = reverse t2 ++ concat a2 (reverse s2)   : concat_eq_append
                         ...    = reverse t2 ++ reverse (a2 :: s2)       : rfl
 
-theorem reverse_reverse : ∀ (l : list T), reverse (reverse l) = l
+theorem reverse_reverse [rewrite] : ∀ (l : list T), reverse (reverse l) = l
 | []       := rfl
 | (a :: l) := calc
     reverse (reverse (a :: l)) = reverse (concat a (reverse l))     : rfl
@@ -178,9 +178,9 @@ definition head [h : inhabited T] : list T → T
 | []       := arbitrary T
 | (a :: l) := a
 
-theorem head_cons [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
+theorem head_cons [rewrite] [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
 
-theorem head_append [h : inhabited T] (t : list T) : ∀ {s : list T}, s ≠ [] → head (s ++ t) = head s
+theorem head_append [rewrite] [h : inhabited T] (t : list T) : ∀ {s : list T}, s ≠ [] → head (s ++ t) = head s
 | []       H := absurd rfl H
 | (a :: s) H :=
   show head (a :: (s ++ t)) = head (a :: s),
@@ -190,9 +190,9 @@ definition tail : list T → list T
 | []       := []
 | (a :: l) := l
 
-theorem tail_nil : tail (@nil T) = []
+theorem tail_nil [rewrite] : tail (@nil T) = []
 
-theorem tail_cons (a : T) (l : list T) : tail (a::l) = l
+theorem tail_cons [rewrite] (a : T) (l : list T) : tail (a::l) = l
 
 theorem cons_head_tail [h : inhabited T] {l : list T} : l ≠ [] → (head l)::(tail l) = l :=
 list.cases_on l
@@ -208,13 +208,13 @@ definition mem : T → list T → Prop
 notation e ∈ s := mem e s
 notation e ∉ s := ¬ e ∈ s
 
-theorem mem_nil_iff (x : T) : x ∈ [] ↔ false :=
+theorem mem_nil_iff [rewrite] (x : T) : x ∈ [] ↔ false :=
 iff.rfl
 
 theorem not_mem_nil (x : T) : x ∉ [] :=
 iff.mp !mem_nil_iff
 
-theorem mem_cons (x : T) (l : list T) : x ∈ x :: l :=
+theorem mem_cons [rewrite] (x : T) (l : list T) : x ∈ x :: l :=
 or.inl rfl
 
 theorem mem_cons_of_mem (y : T) {x : T} {l : list T} : x ∈ l → x ∈ y :: l :=
@@ -340,16 +340,16 @@ definition sublist (l₁ l₂ : list T) := ∀ ⦃a : T⦄, a ∈ l₁ → a ∈
 
 infix `⊆` := sublist
 
-theorem nil_sub (l : list T) : [] ⊆ l :=
+theorem nil_sub [rewrite] (l : list T) : [] ⊆ l :=
 λ b i, false.elim (iff.mp (mem_nil_iff b) i)
 
-theorem sub.refl (l : list T) : l ⊆ l :=
+theorem sub.refl [rewrite] (l : list T) : l ⊆ l :=
 λ b i, i
 
 theorem sub.trans {l₁ l₂ l₃ : list T} (H₁ : l₁ ⊆ l₂) (H₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
 λ b i, H₂ (H₁ i)
 
-theorem sub_cons (a : T) (l : list T) : l ⊆ a::l :=
+theorem sub_cons [rewrite] (a : T) (l : list T) : l ⊆ a::l :=
 λ b i, or.inr i
 
 theorem sub_of_cons_sub {a : T} {l₁ l₂ : list T} : a::l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
@@ -360,10 +360,10 @@ theorem cons_sub_cons  {l₁ l₂ : list T} (a : T) (s : l₁ ⊆ l₂) : (a::l
   (λ e : b = a,  or.inl e)
   (λ i : b ∈ l₁, or.inr (s i))
 
-theorem sub_append_left (l₁ l₂ : list T) : l₁ ⊆ l₁++l₂ :=
+theorem sub_append_left [rewrite] (l₁ l₂ : list T) : l₁ ⊆ l₁++l₂ :=
 λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inl i)
 
-theorem sub_append_right (l₁ l₂ : list T) : l₂ ⊆ l₁++l₂ :=
+theorem sub_append_right [rewrite] (l₁ l₂ : list T) : l₂ ⊆ l₁++l₂ :=
 λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inr i)
 
 theorem sub_cons_of_sub (a : T) {l₁ l₂ : list T} : l₁ ⊆ l₂ → l₁ ⊆ (a::l₂) :=
@@ -399,7 +399,7 @@ definition find : T → list T → nat
 | a []       := 0
 | a (b :: l) := if a = b then 0 else succ (find a l)
 
-theorem find_nil (x : T) : find x [] = 0
+theorem find_nil [rewrite] (x : T) : find x [] = 0
 
 theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
 
@@ -461,9 +461,9 @@ definition nth : list T → nat → option T
 | (a :: l) 0     := some a
 | (a :: l) (n+1) := nth l n
 
-theorem nth_zero (a : T) (l : list T) : nth (a :: l) 0 = some a
+theorem nth_zero [rewrite] (a : T) (l : list T) : nth (a :: l) 0 = some a
 
-theorem nth_succ (a : T) (l : list T) (n : nat) : nth (a::l) (succ n) = nth l n
+theorem nth_succ [rewrite] (a : T) (l : list T) (n : nat) : nth (a::l) (succ n) = nth l n
 
 theorem nth_eq_some : ∀ {l : list T} {n : nat}, n < length l → Σ a : T, nth l n = some a
 | []     n        h := absurd h !not_lt_zero

library/data/nat/basic.lean

@@ -49,10 +49,10 @@ by contradiction
 
 -- add_rewrite succ_ne_zero
 
-theorem pred_zero : pred 0 = 0 :=
+theorem pred_zero [rewrite] : pred 0 = 0 :=
 rfl
 
-theorem pred_succ (n : ℕ) : pred (succ n) = n :=
+theorem pred_succ [rewrite] (n : ℕ) : pred (succ n) = n :=
 rfl
 
 theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
@@ -103,13 +103,13 @@ general m
 
 /- addition -/
 
-theorem add_zero (n : ℕ) : n + 0 = n :=
+theorem add_zero [rewrite] (n : ℕ) : n + 0 = n :=
 rfl
 
-theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
+theorem add_succ [rewrite] (n m : ℕ) : n + succ m = succ (n + m) :=
 rfl
 
-theorem zero_add (n : ℕ) : 0 + n = n :=
+theorem zero_add [rewrite] (n : ℕ) : 0 + n = n :=
 nat.induction_on n
     !add_zero
     (take m IH, show 0 + succ m = succ m, from
@@ -117,7 +117,7 @@ nat.induction_on n
         0 + succ m = succ (0 + m) : add_succ
                ... = succ m       : IH)
 
-theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m) :=
+theorem succ_add [rewrite] (n m : ℕ) : (succ n) + m = succ (n + m) :=
 nat.induction_on m
     (!add_zero ▸ !add_zero)
     (take k IH, calc
@@ -125,7 +125,7 @@ nat.induction_on m
                   ... = succ (succ (n + k))  : IH
                   ... = succ (n + succ k)    : add_succ)
 
-theorem add.comm (n m : ℕ) : n + m = m + n :=
+theorem add.comm [rewrite] (n m : ℕ) : n + m = m + n :=
 nat.induction_on m
     (!add_zero ⬝ !zero_add⁻¹)
     (take k IH, calc
@@ -136,7 +136,7 @@ nat.induction_on m
 theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m :=
 !succ_add ⬝ !add_succ⁻¹
 
-theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
+theorem add.assoc [rewrite] (n m k : ℕ) : (n + m) + k = n + (m + k) :=
 nat.induction_on k
     (!add_zero ▸ !add_zero)
     (take l IH,
@@ -146,7 +146,7 @@ nat.induction_on k
                      ... = n + succ (m + l)    : add_succ
                      ... = n + (m + succ l)    : add_succ)
 
-theorem add.left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) :=
+theorem add.left_comm [rewrite] (n m k : ℕ) : n + (m + k) = m + (n + k) :=
 left_comm add.comm add.assoc n m k
 
 theorem add.right_comm (n m k : ℕ) : n + m + k = n + k + m :=
@@ -186,7 +186,7 @@ eq_zero_of_add_eq_zero_right (!add.comm ⬝ H)
 theorem eq_zero_and_eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
 and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)
 
-theorem add_one (n : ℕ) : n + 1 = succ n :=
+theorem add_one [rewrite] (n : ℕ) : n + 1 = succ n :=
 !add_zero ▸ !add_succ
 
 theorem one_add (n : ℕ) : 1 + n = succ n :=
@@ -194,20 +194,20 @@ theorem one_add (n : ℕ) : 1 + n = succ n :=
 
 /- multiplication -/
 
-theorem mul_zero (n : ℕ) : n * 0 = 0 :=
+theorem mul_zero [rewrite] (n : ℕ) : n * 0 = 0 :=
 rfl
 
-theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
+theorem mul_succ [rewrite] (n m : ℕ) : n * succ m = n * m + n :=
 rfl
 
 -- commutativity, distributivity, associativity, identity
 
-theorem zero_mul (n : ℕ) : 0 * n = 0 :=
+theorem zero_mul [rewrite] (n : ℕ) : 0 * n = 0 :=
 nat.induction_on n
   !mul_zero
   (take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
 
-theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
+theorem succ_mul [rewrite] (n m : ℕ) : (succ n) * m = (n * m) + m :=
 nat.induction_on m
   (!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
   (take k IH, calc
@@ -219,7 +219,7 @@ nat.induction_on m
                 ... = n * k + n + succ k    : add.assoc
                 ... = n * succ k + succ k   : mul_succ)
 
-theorem mul.comm (n m : ℕ) : n * m = m * n :=
+theorem mul.comm [rewrite] (n m : ℕ) : n * m = m * n :=
 nat.induction_on m
   (!mul_zero ⬝ !zero_mul⁻¹)
   (take k IH, calc
@@ -250,7 +250,7 @@ calc
           ... = n * m + k * n  : mul.comm
           ... = n * m + n * k  : mul.comm
 
-theorem mul.assoc (n m k : ℕ) : (n * m) * k = n * (m * k) :=
+theorem mul.assoc [rewrite] (n m k : ℕ) : (n * m) * k = n * (m * k) :=
 nat.induction_on k
   (calc
     (n * m) * 0 = n * (m * 0) : mul_zero)
@@ -261,13 +261,13 @@ nat.induction_on k
                    ... = n * (m * l + m)     : mul.left_distrib
                    ... = n * (m * succ l)    : mul_succ)
 
-theorem mul_one (n : ℕ) : n * 1 = n :=
+theorem mul_one [rewrite] (n : ℕ) : n * 1 = n :=
 calc
   n * 1 = n * 0 + n : mul_succ
     ... = 0 + n     : mul_zero
     ... = n         : zero_add
 
-theorem one_mul (n : ℕ) : 1 * n = n :=
+theorem one_mul [rewrite] (n : ℕ) : 1 * n = n :=
 calc
   1 * n = n * 1 : mul.comm
     ... = n     : mul_one
@@ -314,5 +314,6 @@ section migrate_algebra
   notation a ∣ b := dvd a b
 
   migrate from algebra with nat replacing dvd → dvd
+
 end migrate_algebra
 end nat

library/init/logic.lean

@@ -504,7 +504,7 @@ decidable.rec
   (λ Hnc : ¬c,  eq.refl (@ite c (decidable.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 :=
+theorem if_t_t [rewrite] (c : Prop) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t :=
 decidable.rec
   (λ Hc  : c,  eq.refl (@ite c (decidable.inl Hc)  A t t))
   (λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t t))

library/init/nat.lean

@@ -62,6 +62,12 @@ namespace nat
 
   theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
 
+  theorem le_succ_iff_true [rewrite] (n : ℕ) : n ≤ succ n ↔ true :=
+  iff_true_intro (le_succ n)
+
+  theorem pred_le_iff_true [rewrite] (n : ℕ) : pred n ≤ n ↔ true :=
+  iff_true_intro (pred_le n)
+
   theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
   by induction H2 with n H2 IH;exact H1;exact le.step IH
 
@@ -86,15 +92,24 @@ namespace nat
   theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
   by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
 
+  theorem succ_le_self_iff_false [rewrite] (n : ℕ) : succ n ≤ n ↔ false :=
+  iff_false_intro not_succ_le_self
+
   theorem zero_le (n : ℕ) : 0 ≤ n :=
   by induction n with n IH;apply le.refl;exact le.step IH
 
+  theorem zero_le_iff_true [rewrite] (n : ℕ) : 0 ≤ n ↔ true :=
+  iff_true_intro (zero_le n)
+
   theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
   le.step H
 
   theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
   by induction n with n IH;apply le.refl;exact le.step IH
 
+  theorem zero_lt_succ_iff_true [rewrite] (n : ℕ) : 0 < succ n ↔ true :=
+  iff_true_intro (zero_lt_succ n)
+
   theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
   le.trans (le.step H1) H2
 
@@ -116,9 +131,19 @@ namespace nat
   theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
   by intro H; cases H
 
+  theorem succ_le_zero_iff_false (n : ℕ) : succ n ≤ zero ↔ false :=
+  iff_false_intro (not_succ_le_zero n)
+
   theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
 
+  theorem lt_self_iff_false [rewrite] (n : ℕ) : n < n ↔ false :=
+  iff_false_intro (lt.irrefl n)
+
   theorem self_lt_succ (n : ℕ) : n < succ n := !le.refl
+
+  theorem self_lt_succ_iff_true [rewrite] (n : ℕ) : n < succ n ↔ true :=
+  iff_true_intro (self_lt_succ n)
+
   theorem lt.base (n : ℕ) : n < succ n := !le.refl
 
   theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : false :=
@@ -147,15 +172,18 @@ namespace nat
   theorem lt.trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
   lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
 
-  definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
+  theorem lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
   lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
 
   theorem lt_or_ge (a b : ℕ) : (a < b) ∨ (a ≥ b) :=
   lt_ge_by_cases inl inr
 
-  definition not_lt_zero (a : ℕ) : ¬ a < zero :=
+  theorem not_lt_zero (a : ℕ) : ¬ a < zero :=
   by intro H; cases H
 
+  theorem lt_zero_iff_false [rewrite] (a : ℕ) : a < zero ↔ false :=
+  iff_false_intro (not_lt_zero a)
+
   -- less-than is well-founded
   definition lt.wf [instance] : well_founded lt :=
   begin
@@ -221,7 +249,7 @@ namespace nat
   definition max (a b : ℕ) : ℕ := if a < b then b else a
   definition min (a b : ℕ) : ℕ := if a < b then a else b
 
-  theorem max_self (a : ℕ) : max a a = a :=
+  theorem max_self [rewrite] (a : ℕ) : max a a = a :=
   eq.rec_on !if_t_t rfl
 
   theorem max_eq_right' {a b : ℕ} (H : a < b) : max a b = b :=
@@ -242,6 +270,9 @@ namespace nat
     (λ h : a < b,   le_of_lt (eq.rec_on (eq_max_right h) h))
     (λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
 
+  theorem le_max_left_iff_true [rewrite] (a b : ℕ) : a ≤ max a b ↔ true :=
+  iff_true_intro (le_max_left a b)
+
   theorem le_max_right (a b : ℕ) : b ≤ max a b :=
   by_cases
     (λ h : a < b,   eq.rec_on (eq_max_right h) !le.refl)
@@ -251,13 +282,16 @@ namespace nat
         have aux : a = max a b, from eq_max_left (lt.asymm h),
         eq.rec_on aux (le_of_lt h)))
 
-  theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
+  theorem le_max_right_iff_true [rewrite] (a b : ℕ) : b ≤ max a b ↔ true :=
+  iff_true_intro (le_max_right a b)
+
+  theorem succ_sub_succ_eq_sub [rewrite] (a b : ℕ) : succ a - succ b = a - b :=
   by induction b with b IH;reflexivity; apply congr (eq.refl pred) IH
 
   theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
   eq.rec_on (succ_sub_succ_eq_sub a b) rfl
 
-  theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
+  theorem zero_sub_eq_zero [rewrite] (a : ℕ) : zero - a = zero :=
   nat.rec_on a
     rfl
     (λ a₁ (ih : zero - a₁ = zero), congr (eq.refl pred) ih)
@@ -285,5 +319,12 @@ namespace nat
     (le.refl a)
     (λ b₁ ih, le.trans !pred_le ih)
 
-  lemma sub_lt_succ (a b : ℕ) : a - b < succ a := lt_succ_of_le (sub_le a b)
+  theorem sub_le_iff_true [rewrite] (a b : ℕ) : a - b ≤ a ↔ true :=
+  iff_true_intro (sub_le a b)
+
+  theorem sub_lt_succ (a b : ℕ) : a - b < succ a :=
+  lt_succ_of_le (sub_le a b)
+
+  theorem sub_lt_succ_iff_true [rewrite] (a b : ℕ) : a - b < succ a ↔ true :=
+  iff_true_intro (sub_lt_succ a b)
 end nat

library/logic/connectives.lean

@@ -61,10 +61,10 @@ have Hnp : ¬a, from
   assume Hp : a, absurd (or.inl Hp) not_em,
 absurd (or.inr Hnp) not_em
 
-theorem not_true : ¬ true ↔ false :=
+theorem not_true [rewrite] : ¬ true ↔ false :=
 iff.intro (assume H, H trivial) (assume H, false.elim H)
 
-theorem not_false_iff : ¬ false ↔ true :=
+theorem not_false_iff [rewrite] : ¬ false ↔ true :=
 iff.intro (assume H, trivial) (assume H H1, H1)
 
 theorem not_iff_not (H : a ↔ b) : ¬a ↔ ¬b :=
@@ -105,19 +105,19 @@ iff.intro
    obtain Ha Hb Hc, from H,
    and.intro (and.intro Ha Hb) Hc)
 
-theorem and_true (a : Prop) : a ∧ true ↔ a :=
+theorem and_true [rewrite] (a : Prop) : a ∧ true ↔ a :=
 iff.intro (assume H, and.left H) (assume H, and.intro H trivial)
 
-theorem true_and (a : Prop) : true ∧ a ↔ a :=
+theorem true_and [rewrite] (a : Prop) : true ∧ a ↔ a :=
 iff.intro (assume H, and.right H) (assume H, and.intro trivial H)
 
-theorem and_false (a : Prop) : a ∧ false ↔ false :=
+theorem and_false [rewrite] (a : Prop) : a ∧ false ↔ false :=
 iff.intro (assume H, and.right H) (assume H, false.elim H)
 
-theorem false_and (a : Prop) : false ∧ a ↔ false :=
+theorem false_and [rewrite] (a : Prop) : false ∧ a ↔ false :=
 iff.intro (assume H, and.left H) (assume H, false.elim H)
 
-theorem and_self (a : Prop) : a ∧ a ↔ a :=
+theorem and_self [rewrite] (a : Prop) : a ∧ a ↔ a :=
 iff.intro (assume H, and.left H) (assume H, and.intro H H)
 
 theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
@@ -190,18 +190,18 @@ iff.intro
       (assume Hb, or.inl (or.inr Hb))
       (assume Hc, or.inr Hc)))
 
-theorem or_true (a : Prop) : a ∨ true ↔ true :=
+theorem or_true [rewrite] (a : Prop) : a ∨ true ↔ true :=
 iff.intro (assume H, trivial) (assume H, or.inr H)
 
-theorem true_or (a : Prop) : true ∨ a ↔ true :=
+theorem true_or [rewrite] (a : Prop) : true ∨ a ↔ true :=
 iff.intro (assume H, trivial) (assume H, or.inl H)
 
-theorem or_false (a : Prop) : a ∨ false ↔ a :=
+theorem or_false [rewrite] (a : Prop) : a ∨ false ↔ a :=
 iff.intro
   (assume H, or.elim H (assume H1 : a, H1) (assume H1 : false, false.elim H1))
   (assume H, or.inl H)
 
-theorem false_or (a : Prop) : false ∨ a ↔ a :=
+theorem false_or [rewrite] (a : Prop) : false ∨ a ↔ a :=
 iff.intro
   (assume H, or.elim H (assume H1 : false, false.elim H1) (assume H1 : a, H1))
   (assume H, or.inr H)
@@ -249,22 +249,22 @@ propext (!or.comm) ▸ propext (!or.comm) ▸ propext (!or.comm) ▸ !or.distrib
 definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
 !eq.refl
 
-theorem iff_true (a : Prop) : (a ↔ true) ↔ a :=
+theorem iff_true [rewrite] (a : Prop) : (a ↔ true) ↔ a :=
 iff.intro
   (assume H, iff.mp' H trivial)
   (assume H, iff.intro (assume H1, trivial) (assume H1, H))
 
-theorem true_iff (a : Prop) : (true ↔ a) ↔ a :=
+theorem true_iff [rewrite] (a : Prop) : (true ↔ a) ↔ a :=
 iff.intro
   (assume H, iff.mp H trivial)
   (assume H, iff.intro (assume H1, H) (assume H1, trivial))
 
-theorem iff_false (a : Prop) : (a ↔ false) ↔ ¬ a :=
+theorem iff_false [rewrite] (a : Prop) : (a ↔ false) ↔ ¬ a :=
 iff.intro
   (assume H, and.left H)
   (assume H, and.intro H (assume H1, false.elim H1))
 
-theorem false_iff (a : Prop) : (false ↔ a) ↔ ¬ a :=
+theorem false_iff [rewrite] (a : Prop) : (false ↔ a) ↔ ¬ a :=
 iff.intro
   (assume H, and.right H)
   (assume H, and.intro (assume H1, false.elim H1) H)
@@ -272,7 +272,7 @@ iff.intro
 theorem iff_true_of_self (Ha : a) : a ↔ true :=
 iff.intro (assume H, trivial) (assume H, Ha)
 
-theorem iff_self (a : Prop) : (a ↔ a) ↔ true :=
+theorem iff_self [rewrite] (a : Prop) : (a ↔ a) ↔ true :=
 iff_true_of_self !iff.refl
 
 theorem forall_iff_forall {A : Type} {P Q : A → Prop} (H : ∀a, (P a ↔ Q a)) : (∀a, P a) ↔ ∀a, Q a :=
@@ -291,10 +291,10 @@ section
 
   variables {A : Type} {c₁ c₂ : Prop}
 
-  definition if_true (t e : A) : (if true then t else e) = t :=
+  definition if_true [rewrite] (t e : A) : (if true then t else e) = t :=
   if_pos trivial
 
-  definition if_false (t e : A) : (if false then t else e) = e :=
+  definition if_false [rewrite] (t e : A) : (if false then t else e) = e :=
   if_neg not_false
 end