refactor(library,hott): remove 'by+' and 'begin+'

hott/algebra/ring.hlean

@@ -361,7 +361,7 @@ section
   have b - 1 ≠ 0, from
     suppose b - 1 = 0, H₁ (!zero_add ▸ eq_add_of_sub_eq this),
   have a * b - a = 0, by rewrite H₂; apply sub_self,
-  have a * (b - 1) = 0, by+ rewrite [mul_sub_left_distrib, mul_one]; apply this,
+  have a * (b - 1) = 0, by rewrite [mul_sub_left_distrib, mul_one]; apply this,
     show a = 0, from sum_resolve_left (eq_zero_sum_eq_zero_of_mul_eq_zero this) `b - 1 ≠ 0`
 
   theorem eq_zero_of_mul_eq_self_left {a b : A} (H₁ : b ≠ 1) (H₂ : b * a = a) : a = 0 :=

library/algebra/group_bigops.lean

@@ -244,7 +244,7 @@ namespace finset
       Prod_semigroup dflt (insert a s) f = f a * Prod_semigroup dflt s f :=
     obtain a' (a's : a' ∈ s), from exists_mem_of_ne_empty sne,
     have H : s = insert a' (erase a' s), from eq.symm (insert_erase a's),
-    begin+
+    begin
       rewrite [H, Prod_semigroup_insert_insert dflt f !not_mem_erase (eq.subst H anins)]
     end
   end deceqA
@@ -311,7 +311,7 @@ namespace finset
 
     theorem Prod_singleton (a : A) (f : A → B) : Prod '{a} f = f a :=
     have a ∉ ∅, from not_mem_empty a,
-    by+ rewrite [Prod_insert_of_not_mem f this, Prod_empty, mul_one]
+    by rewrite [Prod_insert_of_not_mem f this, Prod_empty, mul_one]
 
     theorem Prod_image {C : Type} [deceqC : decidable_eq C] {s : finset A} (f : C → B) {g : A → C}
                        (H : set.inj_on g (to_set s)) :
@@ -463,7 +463,7 @@ section Prod
         suppose finite t,
         have finite s, from finite_of_bij_on' H,
         show false, from nfins this,
-      by+ rewrite [Prod_of_not_finite nfins, Prod_of_not_finite nfint])
+      by rewrite [Prod_of_not_finite nfins, Prod_of_not_finite nfint])
 end Prod
 
 /- Sum: sum indexed by a set -/

library/algebra/interval.lean

@@ -127,7 +127,7 @@ open nat eq.ops
 
   proposition Icc_zero (n : ℕ) : '[0, n] = '(-∞, n] :=
   have '[0, n] = '[0, ∞) ∩ '(-∞, n], from rfl,
-  by+ rewrite [this, Ici_zero, univ_inter]
+  by rewrite [this, Ici_zero, univ_inter]
 
   proposition bij_on_add_Icc_zero (m n : ℕ) : bij_on (add m) ('[0, n]) ('[m, m+n]) :=
   have mapsto : ∀₀ i ∈ '[0, n], m + i ∈ '[m, m+n], from

library/algebra/ring.lean

@@ -362,8 +362,8 @@ section
   theorem eq_zero_of_mul_eq_self_right {a b : A} (H₁ : b ≠ 1) (H₂ : a * b = a) : a = 0 :=
   have b - 1 ≠ 0, from
     suppose b - 1 = 0, H₁ (!zero_add ▸ eq_add_of_sub_eq this),
-  have a * b - a = 0,   by  simp,
-  have a * (b - 1) = 0, by+ rewrite [mul_sub_left_distrib, mul_one]; apply this,
+  have a * b - a = 0,   by simp,
+  have a * (b - 1) = 0, by rewrite [mul_sub_left_distrib, mul_one]; apply this,
     show a = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero this) `b - 1 ≠ 0`
 
   theorem eq_zero_of_mul_eq_self_left {a b : A} (H₁ : b ≠ 1) (H₂ : b * a = a) : a = 0 :=

library/data/complex.lean

@@ -260,7 +260,7 @@ protected theorem div_def (z w : ℂ) : z / w = z * w⁻¹ := rfl
 
 theorem of_real_div (x y : ℝ) : of_real (x / y) = of_real x / of_real y :=
 have H : x / y = x * y⁻¹, from rfl,
-by+ rewrite [H, complex.div_def, of_real_mul, of_real_inv]
+by rewrite [H, complex.div_def, of_real_mul, of_real_inv]
 
 theorem conj_inv (z : ℂ) : (conj z)⁻¹ = conj (z⁻¹) :=
 by rewrite [*complex.inv_def, conj_mul, *conj_conj, conj_of_real, cmod_conj]

library/data/list/sorted.lean

@@ -82,7 +82,7 @@ lemma sorted_extends (trans : transitive R) : ∀ {a l}, sorted R (a::l) → all
   have all l (R b),       from sorted_extends (and.right (sorted_inv h)),
   all_of_forall (take x, suppose x ∈ b::l,
     or.elim (eq_or_mem_of_mem_cons this)
-      (suppose x = b, by+ subst x; assumption)
+      (suppose x = b, by subst x; assumption)
       (suppose x ∈ l,
         have R b x, from of_mem_of_all this `all l (R b)`,
         trans `R a b` `R b x`))

library/data/nat/bigops.lean

@@ -96,7 +96,7 @@ proposition sum_range_eq_sum_interval {m n : ℕ} (f : ℕ → A) (H : m ≤ n)
 proposition sum_range_offset (m n : ℕ) (f : ℕ → A) :
             (∑ i = m...m+n, f i) = (∑ i = 0...n, f (m + i)) :=
   have bij_on (add m) ('[0, n]) ('[m, m+n]), from !nat.bij_on_add_Icc_zero,
-  by+ rewrite [-zero_add n at {2}, *sum_range_eq_sum_interval_aux, Sum_eq_of_bij_on f this, zero_add]
+  by rewrite [-zero_add n at {2}, *sum_range_eq_sum_interval_aux, Sum_eq_of_bij_on f this, zero_add]
 
 end set
 
@@ -187,8 +187,8 @@ proposition prod_range_eq_prod_interval {m n : ℕ} (f : ℕ → A) (H : m ≤ n
 proposition prod_range_offset (m n : ℕ) (f : ℕ → A) :
             (∏ i = m...m+n, f i) = (∏ i = 0...n, f (m + i)) :=
   have bij_on (add m) ('[0, n]) ('[m, m+n]), from !nat.bij_on_add_Icc_zero,
-  by+ rewrite [-zero_add n at {2}, *prod_range_eq_prod_interval_aux, Prod_eq_of_bij_on f this,
-               zero_add]
+  by rewrite [-zero_add n at {2}, *prod_range_eq_prod_interval_aux, Prod_eq_of_bij_on f this,
+              zero_add]
 
 end set
 

library/data/rat/basic.lean

@@ -598,7 +598,7 @@ decidable.by_cases
 
 theorem of_nat_div {a b : ℕ} (H : b ∣ a) : of_nat (a / b) = of_nat a / of_nat b :=
 have H' : (int.of_nat b ∣ int.of_nat a), by rewrite [int.of_nat_dvd_of_nat_iff]; exact H,
-by+ rewrite [of_nat_eq, int.of_nat_div, of_int_div H']
+by rewrite [of_nat_eq, int.of_nat_div, of_int_div H']
 
 theorem of_int_pow (a : ℤ) (n : ℕ) : of_int (a^n) = (of_int a)^n :=
 begin

library/data/set/basic.lean

@@ -758,7 +758,7 @@ theorem sInter_singleton (s : set X) : ⋂₀ '{s} = s :=
 ext (take x, iff.intro
   (suppose x ∈ ⋂₀ '{s}, show x ∈ s, from this (mem_singleton s))
   (suppose x ∈ s, take u, suppose u ∈ '{s},
-    show x ∈ u, by+ rewrite [eq_of_mem_singleton this]; assumption))
+    show x ∈ u, by rewrite [eq_of_mem_singleton this]; assumption))
 
 theorem sUnion_union (S T : set (set X)) : ⋃₀ (S ∪ T) = ⋃₀ S ∪ ⋃₀ T :=
 ext (take x, iff.intro

library/data/set/equinumerosity.lean

@@ -116,9 +116,9 @@ open set
   show h a ∈ B, from
     by_cases
       (suppose a ∈ Union U,
-        by+ rewrite [↑h, if_pos this]; exact f_maps_to `a ∈ A`)
+        by rewrite [↑h, if_pos this]; exact f_maps_to `a ∈ A`)
       (suppose a ∉ Union U,
-        by+ rewrite [↑h, if_neg this]; exact ginv_maps_to `a ∈ A`)
+        by rewrite [↑h, if_neg this]; exact ginv_maps_to `a ∈ A`)
 
   /- h is injective -/
 

library/data/set/filter.lean

@@ -409,7 +409,7 @@ le.antisymm
   (le_of_forall_eventually
     (λ P H,
       have P = univ, from eq_of_mem_singleton H,
-      by+ rewrite this; apply eventually_true ⊤))
+      by rewrite this; apply eventually_true ⊤))
   !le_top
 
 theorem sets_bot_eq : sets ⊥ = (univ : set (set A)) :=

library/data/set/function.lean

@@ -104,7 +104,7 @@ take y, assume H,
 obtain x [xmem fxeqy], from H,
 or.elim xmem
   (suppose x ∈ s, or.inl (mem_image this fxeqy))
-  (suppose x ∈ f '- t, or.inr (show y ∈ t, by+ rewrite -fxeqy; exact mem_of_mem_preimage this))
+  (suppose x ∈ f '- t, or.inr (show y ∈ t, by rewrite -fxeqy; exact mem_of_mem_preimage this))
 
 /- maps to -/
 

library/theories/analysis/complex_norm.lean

@@ -43,7 +43,7 @@ namespace complex
     have re x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero H,
     have im x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero
                          (by rewrite [↑ip at H, add.comm at H]; exact H),
-    by+ rewrite [-complex.eta, `re x = 0`, `im x = 0`]
+    by rewrite [-complex.eta, `re x = 0`, `im x = 0`]
   end real_inner_product_space
 
   protected definition real_inner_product_space [reducible] : inner_product_space ℂ :=

library/theories/analysis/inner_product.lean

@@ -114,7 +114,7 @@ end
 private proposition ip_norm_proj_on_eq (u : V) {v : V} (H : v ≠ 0) :
   ip_norm (ip_proj_on u H) = abs ⟨u, v⟩ / ip_norm v :=
 have H1 : ip_norm v ≠ 0, from assume H', H (eq_zero_of_ip_norm_eq_zero H'),
-begin+
+begin
   rewrite [↑ip_proj_on, ip_norm_smul, abs_div, abs_of_nonneg (squared_nonneg (ip_norm v)), pow_two],
   rewrite [div_mul_eq_mul_div, -div_mul_div, div_self H1, mul_one]
 end

library/theories/analysis/metric_space.lean

@@ -120,7 +120,7 @@ exists.intro N
 proposition converges_to_seq_offset_left {X : ℕ → M} {y : M} (k : ℕ) (H : X ⟶ y in ℕ) :
   (λ n, X (k + n)) ⟶ y in ℕ :=
 have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite add.comm),
-by+ rewrite aux; exact converges_to_seq_offset k H
+by rewrite aux; exact converges_to_seq_offset k H
 
 proposition converges_to_seq_offset_succ {X : ℕ → M} {y : M} (H : X ⟶ y in ℕ) :
   (λ n, X (succ n)) ⟶ y in ℕ :=
@@ -142,7 +142,7 @@ proposition converges_to_seq_of_converges_to_seq_offset_left
     {X : ℕ → M} {y : M} {k : ℕ} (H : (λ n, X (k + n)) ⟶ y in ℕ) :
   X ⟶ y in ℕ :=
 have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite add.comm),
-by+ rewrite aux at H; exact converges_to_seq_of_converges_to_seq_offset H
+by rewrite aux at H; exact converges_to_seq_of_converges_to_seq_offset H
 
 proposition converges_to_seq_of_converges_to_seq_offset_succ
     {X : ℕ → M} {y : M} (H : (λ n, X (succ n)) ⟶ y in ℕ) :

library/theories/analysis/normed_space.lean

@@ -74,7 +74,7 @@ namespace analysis
 
   proposition norm_neg (v : V) : ∥ -v ∥ = ∥ v ∥ :=
   have abs (1 : ℝ) = 1, from abs_of_nonneg zero_le_one,
-  by+ rewrite [-@neg_one_smul ℝ V, norm_smul, abs_neg, this, one_mul]
+  by rewrite [-@neg_one_smul ℝ V, norm_smul, abs_neg, this, one_mul]
 
   proposition norm_sub (u v : V) : ∥u - v∥ = ∥v - u∥ :=
     by rewrite [-norm_neg, neg_sub]
@@ -188,7 +188,7 @@ proposition smul_converges_to_seq (c : ℝ) (HX : X ⟶ x in ℕ) :
 by_cases
   (assume cz : c = 0,
     have (λ n, c • X n) = (λ n, 0), from funext (take x, by rewrite [cz, zero_smul]),
-    begin+ rewrite [this, cz, zero_smul], apply converges_to_seq_constant end)
+    begin rewrite [this, cz, zero_smul], apply converges_to_seq_constant end)
   (suppose c ≠ 0, smul_converges_to_seq_aux this HX)
 
 proposition neg_converges_to_seq (HX : X ⟶ x in ℕ) :
@@ -205,7 +205,7 @@ proposition neg_converges_to_seq_iff : ((λ n, - X n) ⟶ - x in ℕ) ↔ (X ⟶
 have aux : X = λ n, (- (- X n)), from funext (take n, by rewrite neg_neg),
 iff.intro
   (assume H : (λ n, -X n)⟶ -x in ℕ,
-    show X ⟶ x in ℕ, by+ rewrite [aux, -neg_neg x]; exact neg_converges_to_seq H)
+    show X ⟶ x in ℕ, by rewrite [aux, -neg_neg x]; exact neg_converges_to_seq H)
   neg_converges_to_seq
 
 proposition norm_converges_to_seq_zero (HX : X ⟶ 0 in ℕ) : (λ n, norm (X n)) ⟶ 0 in ℕ :=
@@ -226,7 +226,7 @@ exists.intro (N : ℕ)
   (take n : ℕ, assume Hn : n ≥ N,
     have HN' : abs (norm (X n) - 0) < ε, from HN n Hn,
     have norm (X n) < ε,
-      by+ rewrite [sub_zero at HN', abs_of_nonneg !norm_nonneg at HN']; apply HN',
+      by rewrite [sub_zero at HN', abs_of_nonneg !norm_nonneg at HN']; apply HN',
     show norm (X n - 0) < ε,
       by rewrite sub_zero; apply this)
 

library/theories/analysis/real_limit.lean

@@ -39,13 +39,13 @@ proposition le_sup {x : ℝ} {X : set ℝ} (Hx : x ∈ X) {b : ℝ} (Hb : ∀ x,
   x ≤ sup X :=
 have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b),
   from and.intro (exists.intro x Hx) (exists.intro b Hb),
-by+ rewrite [↑sup, dif_pos H]; exact and.left (sup_aux_spec H) x Hx
+by rewrite [↑sup, dif_pos H]; exact and.left (sup_aux_spec H) x Hx
 
 proposition sup_le {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → x ≤ b) :
   sup X ≤ b :=
 have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b),
   from and.intro HX (exists.intro b Hb),
-by+ rewrite [↑sup, dif_pos H]; exact and.right (sup_aux_spec H) b Hb
+by rewrite [↑sup, dif_pos H]; exact and.right (sup_aux_spec H) b Hb
 
 proposition exists_mem_and_lt_of_lt_sup {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : b < sup X) :
 ∃ x, x ∈ X ∧ b < x :=
@@ -74,13 +74,13 @@ proposition inf_le {x : ℝ} {X : set ℝ} (Hx : x ∈ X) {b : ℝ} (Hb : ∀ x,
   inf X ≤ x :=
 have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x),
   from and.intro (exists.intro x Hx) (exists.intro b Hb),
-by+ rewrite [↑inf, dif_pos H]; exact and.left (inf_aux_spec H) x Hx
+by rewrite [↑inf, dif_pos H]; exact and.left (inf_aux_spec H) x Hx
 
 proposition le_inf {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → b ≤ x) :
   b ≤ inf X :=
 have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x),
   from and.intro HX (exists.intro b Hb),
-by+ rewrite [↑inf, dif_pos H]; exact and.right (inf_aux_spec H) b Hb
+by rewrite [↑inf, dif_pos H]; exact and.right (inf_aux_spec H) b Hb
 
 proposition exists_mem_and_lt_of_inf_lt {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : inf X < b) :
 ∃ x, x ∈ X ∧ x < b :=
@@ -261,7 +261,7 @@ smul_converges_to_seq c HX
 proposition mul_right_converges_to_seq (c : ℝ) (HX : X ⟶ x in ℕ) :
   (λ n, X n * c) ⟶ x * c in ℕ :=
 have (λ n, X n * c) = (λ n, c * X n), from funext (take x, !mul.comm),
-by+ rewrite [this, mul.comm]; apply mul_left_converges_to_seq c HX
+by rewrite [this, mul.comm]; apply mul_left_converges_to_seq c HX
 
 theorem converges_to_seq_squeeze (HX : X ⟶ x in ℕ) (HY : Y ⟶ x in ℕ) {Z : ℕ → ℝ} (HZX : ∀ n, X n ≤ Z n)
         (HZY : ∀ n, Z n ≤ Y n) : Z ⟶ x in ℕ :=
@@ -429,7 +429,7 @@ have ∀ n, X i ≤ X (i + n), from
     (by rewrite nat.add_zero; apply le.refl)
     (take n, assume ih, le.trans ih (H (i + n))),
 have X i ≤ X (i + (j - i)), from !this,
-by+ rewrite [add_sub_of_le `i ≤ j` at this]; exact this
+by rewrite [add_sub_of_le `i ≤ j` at this]; exact this
 
 proposition converges_to_seq_sup_of_nondecreasing (nondecX : nondecreasing X) {b : ℝ}
     (Hb : ∀ i, X i ≤ b) : X ⟶ sup (X ' univ) in ℕ :=
@@ -505,7 +505,7 @@ have H₃ : {x : ℝ | -x ∈ X ' univ} = {x : ℝ | x ∈ (λ n, -X n) ' univ},
   {x : ℝ | -x ∈ X ' univ} = (λ y, -y) ' (X ' univ) : by rewrite image_neg_eq
                        ... = {x : ℝ | x ∈ (λ n, -X n) ' univ} : image_comp,
 have H₄ : ∀ i, - X i ≤ - b, from take i, neg_le_neg (Hb i),
-begin+
+begin
   -- need krewrite here
   krewrite [-neg_converges_to_seq_iff, -sup_neg H₁ H₂, H₃, -nondecreasing_neg_iff at nonincX],
   apply converges_to_seq_sup_of_nondecreasing nonincX H₄

library/theories/group_theory/cyclic.lean

@@ -232,7 +232,7 @@ end
 definition pow_fin_is_iso (a : A) : is_iso_class (pow_fin' a) :=
 is_iso_class.mk (pow_fin_hom a)
   (have H : injective (λ (i : fin (order a)), a ^ (val i + 0)), from pow_fin_inj a 0,
-    begin+ rewrite [↑pow_fin', succ_pred_of_pos !order_pos]; exact H end)
+    begin rewrite [↑pow_fin', succ_pred_of_pos !order_pos]; exact H end)
 
 end cyclic
 

library/theories/measure_theory/extended_real.lean

@@ -161,7 +161,7 @@ protected theorem neg_zero : -(0 : ereal) = 0 := rfl
 
 theorem infty_mul_pos {x : real} (H : x > 0) : ∞ * x = ∞ :=
 have H1 : x ≠ 0, from ne_of_gt H,
-by+ rewrite [*ereal.mul_def, ↑ereal.mul, if_neg H1, if_pos H]
+by rewrite [*ereal.mul_def, ↑ereal.mul, if_neg H1, if_pos H]
 
 theorem pos_mul_infty {x : real} (H : x > 0) : x * ∞ = ∞ :=
 by rewrite [ereal.mul_comm, infty_mul_pos H]
@@ -169,7 +169,7 @@ by rewrite [ereal.mul_comm, infty_mul_pos H]
 theorem infty_mul_neg {x : real} (H : x < 0) : ∞ * x = -∞ :=
 have H1 : x ≠ 0, from ne_of_lt H,
 have H2 : ¬ x > 0, from not_lt_of_gt H,
-by+ rewrite [*ereal.mul_def, ↑ereal.mul, if_neg H1, if_neg H2]
+by rewrite [*ereal.mul_def, ↑ereal.mul, if_neg H1, if_neg H2]
 
 theorem neg_mul_infty {x : real} (H : x < 0) : x * ∞ = -∞ :=
 by rewrite [ereal.mul_comm, infty_mul_neg H]