chore: deprecate Ordinal.natCast_succ → Nat.cast_add_one (leanprover-community#39643)

The eventual goal is to write `x + 1` instead of `succ x`.

Author: vihdzp

Mathlib/SetTheory/Cardinal/Aleph.lean

@@ -600,9 +600,7 @@ theorem isNormal_preBeth : Order.IsNormal preBeth := by
 
 theorem preBeth_nat : ∀ n : ℕ, preBeth n = (2 ^ ·)^[n] (0 : ℕ)
   | 0 => by simp
-  | n + 1 => by
-    rw [natCast_succ, preBeth_succ, Function.iterate_succ_apply', preBeth_nat]
-    simp
+  | n + 1 => by simp [Function.iterate_succ_apply', preBeth_nat]
 
 @[simp]
 theorem preBeth_one : preBeth 1 = 1 := by

Mathlib/SetTheory/Ordinal/Basic.lean

@@ -953,8 +953,9 @@ protected theorem le_one_iff {a : Ordinal} : a ≤ 1 ↔ a = 0 ∨ a = 1 :=
 theorem card_succ (o : Ordinal) : card (succ o) = card o + 1 := by
   simp
 
+@[deprecated Nat.cast_add_one (since := "2026-05-21")]
 theorem natCast_succ (n : ℕ) : ↑n.succ = succ (n : Ordinal) :=
-  rfl
+  n.cast_add_one
 
 instance uniqueIioOne : Unique (Iio (1 : Ordinal)) where
   default := ⟨0, zero_lt_one' Ordinal⟩

Mathlib/SetTheory/Ordinal/Exponential.lean

@@ -490,8 +490,8 @@ theorem lt_omega0_opow {a b : Ordinal} (hb : b ≠ 0) :
   refine ⟨fun ha ↦ ⟨_, lt_log_of_lt_opow hb ha, ?_⟩,
     fun ⟨c, hc, n, hn⟩ ↦ hn.trans (omega0_opow_mul_nat_lt hc n)⟩
   obtain ⟨n, hn⟩ := lt_omega0.1 (div_opow_log_lt a one_lt_omega0)
-  use n.succ
-  rw [natCast_succ, ← hn]
+  use n + 1
+  rw [Nat.cast_add_one, ← hn]
   exact lt_mul_succ_div a (opow_ne_zero _ omega0_ne_zero)
 
 theorem lt_omega0_opow_succ {a b : Ordinal} : a < ω ^ succ b ↔ ∃ n : ℕ, a < ω ^ b * n := by

Mathlib/SetTheory/Ordinal/Notation.lean

@@ -557,7 +557,7 @@ theorem repr_mul : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ * o₂) = rep
     · obtain ⟨x, xe⟩ := Nat.exists_eq_succ_of_ne_zero n₂.ne_zero
       simp only [Mul.mul, mul, e0, ↓reduceIte, repr, PNat.mul_coe, natCast_mul, opow_zero, one_mul]
       simp only [xe, h₂.zero_of_zero e0, repr, add_zero]
-      rw [natCast_succ x, add_mul_succ _ ao, mul_assoc]
+      rw [Nat.cast_add_one x, ← succ_eq_add_one, add_mul_succ _ ao, mul_assoc]
     · simp only [repr]
       haveI := h₁.fst
       haveI := h₂.fst
@@ -842,9 +842,10 @@ theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω 
   calc
     (ω0 ^ (k.succ : Ordinal)) * α' + R'
     _ = (ω0 ^ succ (k : Ordinal)) * α' + ((ω0 ^ (k : Ordinal)) * α' * m + R) := by
-        rw [natCast_succ, RR, ← mul_assoc]
+        rw [Nat.cast_add_one, RR, ← mul_assoc, succ_eq_add_one]
     _ = ((ω0 ^ (k : Ordinal)) * α' + R) * α' + ((ω0 ^ (k : Ordinal)) * α' + R) * m := ?_
-    _ = (α' + m) ^ succ (k.succ : Ordinal) := by rw [← mul_add, natCast_succ, opow_succ, IH.2]
+    _ = (α' + m) ^ succ (k.succ : Ordinal) := by
+        rw [← mul_add, opow_succ, Nat.cast_add_one, IH.2, succ_eq_add_one]
   congr 1
   · have αd : ω ∣ α' :=
       dvd_add (dvd_mul_of_dvd_left (by simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0)) _) d
@@ -865,7 +866,7 @@ theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω 
   · cases m
     · have : R = 0 := by cases k <;> simp [R, opowAux]
       simp [this]
-    · rw [natCast_succ, add_mul_succ]
+    · rw [Nat.cast_add_one, ← succ_eq_add_one, add_mul_succ]
       apply add_of_omega0_opow_le Rl
       rw [opow_mul, opow_succ]
       gcongr
@@ -1010,14 +1011,14 @@ theorem fundamentalSequence_has_prop (o) : FundamentalSequenceProp o (fundamenta
       refine
         ⟨isSuccLimit_mul_right this isSuccLimit_omega0, fun i =>
           ⟨this, ?_, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)⟩, exists_lt_mul_omega0'⟩
-      rw [← mul_succ, ← natCast_succ]
+      rw [← mul_add_one, ← Nat.cast_add_one]
       gcongr
       apply natCast_lt_omega0
     · have := opow_pos (repr a') omega0_pos
       refine
         ⟨isSuccLimit_add _ (isSuccLimit_mul_right this isSuccLimit_omega0), fun i => ⟨this, ?_, ?_⟩,
           exists_lt_add exists_lt_mul_omega0'⟩
-      · rw [← mul_succ, ← natCast_succ]
+      · rw [← mul_add_one, ← Nat.cast_add_one]
         gcongr
         apply natCast_lt_omega0
       · refine fun H => H.fst.oadd _ (NF.below_of_lt' ?_ (@NF.oadd_zero _ _ (iha.2 H.fst)))