refactor(library/data/nat/basic): rename pred.zero and pred.succ

library/data/nat/basic.lean

@@ -53,17 +53,17 @@ assume H, no_confusion H
 
 -- add_rewrite succ_ne_zero
 
-theorem pred.zero : pred 0 = 0 :=
+theorem pred_zero : pred 0 = 0 :=
 rfl
 
-theorem pred.succ (n : ℕ) : pred (succ n) = n :=
+theorem pred_succ (n : ℕ) : pred (succ n) = n :=
 rfl
 
 theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
 induction_on n
   (or.inl rfl)
   (take m IH, or.inr
-    (show succ m = succ (pred (succ m)), from congr_arg succ !pred.succ⁻¹))
+    (show succ m = succ (pred (succ m)), from congr_arg succ !pred_succ⁻¹))
 
 theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃k : ℕ, n = succ k :=
 exists.intro _ (or_resolve_right !eq_zero_or_eq_succ_pred H)

library/data/nat/order.lean

@@ -285,16 +285,16 @@ nat.cases_on n
   (assume H : pred 0 ≤ m, !zero_le)
   (take n',
     assume H : pred (succ n') ≤ m,
-    have H1 : n' ≤ m, from pred.succ n' ▸ H,
+    have H1 : n' ≤ m, from pred_succ n' ▸ H,
     succ_le_succ H1)
 
 theorem pred_le_of_le_succ {n m : ℕ} : n ≤ succ m → pred n ≤ m :=
 nat.cases_on n
-  (assume H, !pred.zero⁻¹ ▸ zero_le m)
+  (assume H, !pred_zero⁻¹ ▸ zero_le m)
   (take n',
     assume H : succ n' ≤ succ m,
     have H1 : n' ≤ m, from le_of_succ_le_succ H,
-    !pred.succ⁻¹ ▸ H1)
+    !pred_succ⁻¹ ▸ H1)
 
 theorem succ_le_of_le_pred {n m : ℕ} : succ n ≤ m → n ≤ pred m :=
 nat.cases_on m
@@ -302,14 +302,14 @@ nat.cases_on m
   (take m',
     assume H : succ n ≤ succ m',
     have H1 : n ≤ m', from le_of_succ_le_succ H,
-    !pred.succ⁻¹ ▸ H1)
+    !pred_succ⁻¹ ▸ H1)
 
 theorem pred_le_pred_of_le {n m : ℕ} : n ≤ m → pred n ≤ pred m :=
 nat.cases_on n
-  (assume H, pred.zero⁻¹ ▸ zero_le (pred m))
+  (assume H, pred_zero⁻¹ ▸ zero_le (pred m))
   (take n',
     assume H : succ n' ≤ m,
-    !pred.succ⁻¹ ▸ succ_le_of_le_pred H)
+    !pred_succ⁻¹ ▸ succ_le_of_le_pred H)
 
 theorem le_or_eq_succ_of_le_succ {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m :=
 or_of_or_of_imp_left (succ_le_or_eq_of_le H)
@@ -317,8 +317,8 @@ or_of_or_of_imp_left (succ_le_or_eq_of_le H)
 
 theorem le_pred_self (n : ℕ) : pred n ≤ n :=
 cases_on n
-  (pred.zero⁻¹ ▸ !le.refl)
-  (take k : ℕ, (!pred.succ)⁻¹ ▸ !self_le_succ)
+  (pred_zero⁻¹ ▸ !le.refl)
+  (take k : ℕ, (!pred_succ)⁻¹ ▸ !self_le_succ)
 
 theorem succ_pos (n : ℕ) : 0 < succ n :=
 !zero_lt_succ

library/data/nat/sub.lean

@@ -33,7 +33,7 @@ induction_on n !sub_zero_right
     calc
       0 - succ k = pred (0 - k) : !sub_succ_right
              ... = pred 0       : {IH}
-             ... = 0            : pred.zero)
+             ... = 0            : pred_zero)
 
 theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m :=
 succ_sub_succ_eq_sub n m
@@ -118,14 +118,14 @@ theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
 theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m :=
 induction_on n
   (calc
-    pred 0 * m = 0 * m     : {pred.zero}
+    pred 0 * m = 0 * m     : {pred_zero}
            ... = 0         : !zero_mul
            ... = 0 - m     : !sub_zero_left⁻¹
            ... = 0 * m - m : {!zero_mul⁻¹})
   (take k : nat,
     assume IH : pred k * m = k * m - m,
     calc
-      pred (succ k) * m = k * m          : {!pred.succ}
+      pred (succ k) * m = k * m          : {!pred_succ}
                     ... = k * m + m - m  : !sub_add_left⁻¹
                     ... = succ k * m - m : {!succ_mul⁻¹})