chore: cleanup imports of Array.Lemmas (#5246)
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Kim Morrison
GitHub
parent
318e455d96
commit
744b68358e
5 changed files with 176 additions and 171 deletions
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@ -4,11 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Mario Carneiro
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-/
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prelude
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import Init.Data.Nat.MinMax
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import Init.Data.Nat.Lemmas
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import Init.Data.List.Monadic
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import Init.Data.List.Nat.Range
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import Init.Data.Fin.Basic
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import Init.Data.List.Range
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import Init.Data.Array.Mem
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import Init.TacticsExtra
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@ -22,56 +22,11 @@ open Nat
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/-! ### range' -/
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theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step) n step := by
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simp [range', Nat.add_succ, Nat.mul_succ]
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@[simp] theorem length_range' (s step) : ∀ n : Nat, length (range' s n step) = n
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| 0 => rfl
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| _ + 1 => congrArg succ (length_range' _ _ _)
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@[simp] theorem range'_eq_nil : range' s n step = [] ↔ n = 0 := by
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rw [← length_eq_zero, length_range']
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theorem range'_ne_nil (s n : Nat) : range' s n ≠ [] ↔ n ≠ 0 := by
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cases n <;> simp
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@[simp] theorem range'_zero : range' s 0 = [] := by
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simp
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@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
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@[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
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constructor
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· intro h
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have h' := congrArg List.length h
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simp at h'
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subst h'
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cases n with
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| zero => simp
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| succ n =>
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simp only [range'_succ] at h
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simp_all
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· rintro ⟨rfl, rfl | rfl⟩ <;> simp
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theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
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| 0 => by simp [range', Nat.not_lt_zero]
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| n + 1 => by
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have h (i) : i ≤ n ↔ i = 0 ∨ ∃ j, i = succ j ∧ j < n := by
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cases i <;> simp [Nat.succ_le, Nat.succ_inj']
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simp [range', mem_range', Nat.lt_succ, h]; simp only [← exists_and_right, and_assoc]
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rw [exists_comm]; simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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@[simp] theorem mem_range'_1 : m ∈ range' s n ↔ s ≤ m ∧ m < s + n := by
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simp [mem_range']; exact ⟨
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fun ⟨i, h, e⟩ => e ▸ ⟨Nat.le_add_right .., Nat.add_lt_add_left h _⟩,
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fun ⟨h₁, h₂⟩ => ⟨m - s, Nat.sub_lt_left_of_lt_add h₁ h₂, (Nat.add_sub_cancel' h₁).symm⟩⟩
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theorem head?_range' (n : Nat) : (range' s n).head? = if n = 0 then none else some s := by
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induction n <;> simp_all [range'_succ, head?_append]
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@[simp] theorem head_range' (n : Nat) (h) : (range' s n).head h = s := by
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repeat simp_all [head?_range', head_eq_iff_head?_eq_some]
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theorem getLast?_range' (n : Nat) : (range' s n).getLast? = if n = 0 then none else some (s + n - 1) := by
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induction n generalizing s with
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| zero => simp
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@ -116,67 +71,12 @@ theorem pairwise_le_range' s n (step := 1) :
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theorem nodup_range' (s n : Nat) (step := 1) (h : 0 < step := by simp) : Nodup (range' s n step) :=
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(pairwise_lt_range' s n step h).imp Nat.ne_of_lt
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@[simp]
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theorem map_add_range' (a) : ∀ s n step, map (a + ·) (range' s n step) = range' (a + s) n step
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| _, 0, _ => rfl
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| s, n + 1, step => by simp [range', map_add_range' _ (s + step) n step, Nat.add_assoc]
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theorem map_sub_range' (a s n : Nat) (h : a ≤ s) :
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map (· - a) (range' s n step) = range' (s - a) n step := by
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conv => lhs; rw [← Nat.add_sub_cancel' h]
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rw [← map_add_range', map_map, (?_ : _∘_ = _), map_id]
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funext x; apply Nat.add_sub_cancel_left
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theorem range'_append : ∀ s m n step : Nat,
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range' s m step ++ range' (s + step * m) n step = range' s (n + m) step
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| s, 0, n, step => rfl
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| s, m + 1, n, step => by
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simpa [range', Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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using range'_append (s + step) m n step
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@[simp] theorem range'_append_1 (s m n : Nat) :
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range' s m ++ range' (s + m) n = range' s (n + m) := by simpa using range'_append s m n 1
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theorem range'_sublist_right {s m n : Nat} : range' s m step <+ range' s n step ↔ m ≤ n :=
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⟨fun h => by simpa only [length_range'] using h.length_le,
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fun h => by rw [← Nat.sub_add_cancel h, ← range'_append]; apply sublist_append_left⟩
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theorem range'_subset_right {s m n : Nat} (step0 : 0 < step) :
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range' s m step ⊆ range' s n step ↔ m ≤ n := by
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refine ⟨fun h => Nat.le_of_not_lt fun hn => ?_, fun h => (range'_sublist_right.2 h).subset⟩
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have ⟨i, h', e⟩ := mem_range'.1 <| h <| mem_range'.2 ⟨_, hn, rfl⟩
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exact Nat.ne_of_gt h' (Nat.eq_of_mul_eq_mul_left step0 (Nat.add_left_cancel e))
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theorem range'_subset_right_1 {s m n : Nat} : range' s m ⊆ range' s n ↔ m ≤ n :=
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range'_subset_right (by decide)
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theorem getElem?_range' (s step) :
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∀ {m n : Nat}, m < n → (range' s n step)[m]? = some (s + step * m)
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| 0, n + 1, _ => by simp [range'_succ]
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| m + 1, n + 1, h => by
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simp only [range'_succ, getElem?_cons_succ]
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exact (getElem?_range' (s + step) step (Nat.lt_of_add_lt_add_right h)).trans <| by
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simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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@[simp] theorem getElem_range' {n m step} (i) (H : i < (range' n m step).length) :
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(range' n m step)[i] = n + step * i :=
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(getElem?_eq_some.1 <| getElem?_range' n step (by simpa using H)).2
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theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [s + step * n] := by
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rw [Nat.add_comm n 1]; exact (range'_append s n 1 step).symm
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theorem range'_1_concat (s n : Nat) : range' s (n + 1) = range' s n ++ [s + n] := by
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simp [range'_concat]
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theorem range'_eq_cons_iff : range' s n = a :: xs ↔ s = a ∧ 0 < n ∧ xs = range' (a + 1) (n - 1) := by
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induction n generalizing s with
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| zero => simp
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| succ n ih =>
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simp only [range'_succ]
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simp only [cons.injEq, and_congr_right_iff]
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rintro rfl
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simp [eq_comm]
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@[simp] theorem range'_eq_singleton {s n a : Nat} : range' s n = [a] ↔ s = a ∧ n = 1 := by
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rw [range'_eq_cons_iff]
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simp only [nil_eq, range'_eq_nil, and_congr_right_iff]
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@ -253,17 +153,6 @@ theorem erase_range' :
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/-! ### range -/
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theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0 (n + s)
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| 0, n => rfl
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| s + 1, n => by rw [← Nat.add_assoc, Nat.add_right_comm n s 1]; exact range_loop_range' s (n + 1)
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theorem range_eq_range' (n : Nat) : range n = range' 0 n :=
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(range_loop_range' n 0).trans <| by rw [Nat.zero_add]
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theorem range_succ_eq_map (n : Nat) : range (n + 1) = 0 :: map succ (range n) := by
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rw [range_eq_range', range_eq_range', range', Nat.add_comm, ← map_add_range']
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congr; exact funext (Nat.add_comm 1)
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theorem reverse_range' : ∀ s n : Nat, reverse (range' s n) = map (s + n - 1 - ·) (range n)
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| s, 0 => rfl
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| s, n + 1 => by
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@ -271,26 +160,6 @@ theorem reverse_range' : ∀ s n : Nat, reverse (range' s n) = map (s + n - 1 -
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show s + (n + 1) - 1 = s + n from rfl, map, map_map]
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simp [reverse_range', Nat.sub_right_comm, Nat.sub_sub]
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theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) := by
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rw [range_eq_range', map_add_range']; rfl
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@[simp] theorem length_range (n : Nat) : length (range n) = n := by
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simp only [range_eq_range', length_range']
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@[simp] theorem range_eq_nil {n : Nat} : range n = [] ↔ n = 0 := by
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rw [← length_eq_zero, length_range]
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theorem range_ne_nil (n : Nat) : range n ≠ [] ↔ n ≠ 0 := by
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cases n <;> simp
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@[simp]
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theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
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simp only [range_eq_range', range'_sublist_right]
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@[simp]
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theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by
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simp only [range_eq_range', range'_subset_right, lt_succ_self]
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@[simp]
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theorem mem_range {m n : Nat} : m ∈ range n ↔ m < n := by
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simp only [range_eq_range', mem_range'_1, Nat.zero_le, true_and, Nat.zero_add]
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@ -305,43 +174,6 @@ theorem pairwise_lt_range (n : Nat) : Pairwise (· < ·) (range n) := by
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theorem pairwise_le_range (n : Nat) : Pairwise (· ≤ ·) (range n) :=
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Pairwise.imp Nat.le_of_lt (pairwise_lt_range _)
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theorem getElem?_range {m n : Nat} (h : m < n) : (range n)[m]? = some m := by
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simp [range_eq_range', getElem?_range' _ _ h]
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@[simp] theorem getElem_range {n : Nat} (m) (h : m < (range n).length) : (range n)[m] = m := by
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simp [range_eq_range']
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theorem range_succ (n : Nat) : range (succ n) = range n ++ [n] := by
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simp only [range_eq_range', range'_1_concat, Nat.zero_add]
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theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·) := by
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rw [← range'_eq_map_range]
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simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
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theorem head?_range (n : Nat) : (range n).head? = if n = 0 then none else some 0 := by
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induction n with
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| zero => simp
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| succ n ih =>
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simp only [range_succ, head?_append, ih]
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split <;> simp_all
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@[simp] theorem head_range (n : Nat) (h) : (range n).head h = 0 := by
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cases n with
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| zero => simp at h
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| succ n => simp [head?_range, head_eq_iff_head?_eq_some]
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theorem getLast?_range (n : Nat) : (range n).getLast? = if n = 0 then none else some (n - 1) := by
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induction n with
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| zero => simp
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| succ n ih =>
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simp only [range_succ, getLast?_append, ih]
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split <;> simp_all
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@[simp] theorem getLast_range (n : Nat) (h) : (range n).getLast h = n - 1 := by
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cases n with
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| zero => simp at h
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| succ n => simp [getLast?_range, getLast_eq_iff_getLast_eq_some]
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theorem take_range (m n : Nat) : take m (range n) = range (min m n) := by
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apply List.ext_getElem
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· simp
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@ -19,6 +19,179 @@ open Nat
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/-! ## Ranges and enumeration -/
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/-! ### range' -/
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theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step) n step := by
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simp [range', Nat.add_succ, Nat.mul_succ]
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@[simp] theorem length_range' (s step) : ∀ n : Nat, length (range' s n step) = n
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| 0 => rfl
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| _ + 1 => congrArg succ (length_range' _ _ _)
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@[simp] theorem range'_eq_nil : range' s n step = [] ↔ n = 0 := by
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rw [← length_eq_zero, length_range']
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theorem range'_ne_nil (s n : Nat) : range' s n ≠ [] ↔ n ≠ 0 := by
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cases n <;> simp
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@[simp] theorem range'_zero : range' s 0 = [] := by
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simp
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@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
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@[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
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constructor
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· intro h
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have h' := congrArg List.length h
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simp at h'
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subst h'
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cases n with
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| zero => simp
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| succ n =>
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simp only [range'_succ] at h
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simp_all
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· rintro ⟨rfl, rfl | rfl⟩ <;> simp
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theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
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| 0 => by simp [range', Nat.not_lt_zero]
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| n + 1 => by
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have h (i) : i ≤ n ↔ i = 0 ∨ ∃ j, i = succ j ∧ j < n := by
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cases i <;> simp [Nat.succ_le, Nat.succ_inj']
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simp [range', mem_range', Nat.lt_succ, h]; simp only [← exists_and_right, and_assoc]
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rw [exists_comm]; simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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theorem getElem?_range' (s step) :
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∀ {m n : Nat}, m < n → (range' s n step)[m]? = some (s + step * m)
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| 0, n + 1, _ => by simp [range'_succ]
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| m + 1, n + 1, h => by
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simp only [range'_succ, getElem?_cons_succ]
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exact (getElem?_range' (s + step) step (by exact succ_lt_succ_iff.mp h)).trans <| by
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simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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@[simp] theorem getElem_range' {n m step} (i) (H : i < (range' n m step).length) :
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(range' n m step)[i] = n + step * i :=
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(getElem?_eq_some.1 <| getElem?_range' n step (by simpa using H)).2
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theorem head?_range' (n : Nat) : (range' s n).head? = if n = 0 then none else some s := by
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induction n <;> simp_all [range'_succ, head?_append]
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@[simp] theorem head_range' (n : Nat) (h) : (range' s n).head h = s := by
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repeat simp_all [head?_range', head_eq_iff_head?_eq_some]
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@[simp]
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theorem map_add_range' (a) : ∀ s n step, map (a + ·) (range' s n step) = range' (a + s) n step
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| _, 0, _ => rfl
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| s, n + 1, step => by simp [range', map_add_range' _ (s + step) n step, Nat.add_assoc]
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theorem range'_append : ∀ s m n step : Nat,
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range' s m step ++ range' (s + step * m) n step = range' s (n + m) step
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| s, 0, n, step => rfl
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| s, m + 1, n, step => by
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simpa [range', Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
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using range'_append (s + step) m n step
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@[simp] theorem range'_append_1 (s m n : Nat) :
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range' s m ++ range' (s + m) n = range' s (n + m) := by simpa using range'_append s m n 1
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theorem range'_sublist_right {s m n : Nat} : range' s m step <+ range' s n step ↔ m ≤ n :=
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⟨fun h => by simpa only [length_range'] using h.length_le,
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fun h => by rw [← Nat.sub_add_cancel h, ← range'_append]; apply sublist_append_left⟩
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theorem range'_subset_right {s m n : Nat} (step0 : 0 < step) :
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range' s m step ⊆ range' s n step ↔ m ≤ n := by
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refine ⟨fun h => Nat.le_of_not_lt fun hn => ?_, fun h => (range'_sublist_right.2 h).subset⟩
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have ⟨i, h', e⟩ := mem_range'.1 <| h <| mem_range'.2 ⟨_, hn, rfl⟩
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exact Nat.ne_of_gt h' (Nat.eq_of_mul_eq_mul_left step0 (Nat.add_left_cancel e))
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theorem range'_subset_right_1 {s m n : Nat} : range' s m ⊆ range' s n ↔ m ≤ n :=
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range'_subset_right (by decide)
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theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [s + step * n] := by
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rw [Nat.add_comm n 1]; exact (range'_append s n 1 step).symm
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theorem range'_1_concat (s n : Nat) : range' s (n + 1) = range' s n ++ [s + n] := by
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simp [range'_concat]
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theorem range'_eq_cons_iff : range' s n = a :: xs ↔ s = a ∧ 0 < n ∧ xs = range' (a + 1) (n - 1) := by
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induction n generalizing s with
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| zero => simp
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| succ n ih =>
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simp only [range'_succ]
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simp only [cons.injEq, and_congr_right_iff]
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rintro rfl
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simp [eq_comm]
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/-! ### range -/
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theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0 (n + s)
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| 0, n => rfl
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| s + 1, n => by rw [← Nat.add_assoc, Nat.add_right_comm n s 1]; exact range_loop_range' s (n + 1)
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theorem range_eq_range' (n : Nat) : range n = range' 0 n :=
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(range_loop_range' n 0).trans <| by rw [Nat.zero_add]
|
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|
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theorem getElem?_range {m n : Nat} (h : m < n) : (range n)[m]? = some m := by
|
||||
simp [range_eq_range', getElem?_range' _ _ h]
|
||||
|
||||
@[simp] theorem getElem_range {n : Nat} (m) (h : m < (range n).length) : (range n)[m] = m := by
|
||||
simp [range_eq_range']
|
||||
|
||||
theorem range_succ_eq_map (n : Nat) : range (n + 1) = 0 :: map succ (range n) := by
|
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rw [range_eq_range', range_eq_range', range', Nat.add_comm, ← map_add_range']
|
||||
congr; exact funext (Nat.add_comm 1)
|
||||
|
||||
theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) := by
|
||||
rw [range_eq_range', map_add_range']; rfl
|
||||
|
||||
@[simp] theorem length_range (n : Nat) : length (range n) = n := by
|
||||
simp only [range_eq_range', length_range']
|
||||
|
||||
@[simp] theorem range_eq_nil {n : Nat} : range n = [] ↔ n = 0 := by
|
||||
rw [← length_eq_zero, length_range]
|
||||
|
||||
theorem range_ne_nil (n : Nat) : range n ≠ [] ↔ n ≠ 0 := by
|
||||
cases n <;> simp
|
||||
|
||||
@[simp]
|
||||
theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
|
||||
simp only [range_eq_range', range'_sublist_right]
|
||||
|
||||
@[simp]
|
||||
theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by
|
||||
simp only [range_eq_range', range'_subset_right, lt_succ_self]
|
||||
|
||||
theorem range_succ (n : Nat) : range (succ n) = range n ++ [n] := by
|
||||
simp only [range_eq_range', range'_1_concat, Nat.zero_add]
|
||||
|
||||
theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·) := by
|
||||
rw [← range'_eq_map_range]
|
||||
simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
|
||||
|
||||
theorem head?_range (n : Nat) : (range n).head? = if n = 0 then none else some 0 := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih =>
|
||||
simp only [range_succ, head?_append, ih]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem head_range (n : Nat) (h) : (range n).head h = 0 := by
|
||||
cases n with
|
||||
| zero => simp at h
|
||||
| succ n => simp [head?_range, head_eq_iff_head?_eq_some]
|
||||
|
||||
theorem getLast?_range (n : Nat) : (range n).getLast? = if n = 0 then none else some (n - 1) := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih =>
|
||||
simp only [range_succ, getLast?_append, ih]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem getLast_range (n : Nat) (h) : (range n).getLast h = n - 1 := by
|
||||
cases n with
|
||||
| zero => simp at h
|
||||
| succ n => simp [getLast?_range, getLast_eq_iff_getLast_eq_some]
|
||||
|
||||
/-! ### enumFrom -/
|
||||
|
||||
@[simp]
|
||||
|
|
|
|||
|
|
@ -5,6 +5,7 @@ Authors: Kim Morrison
|
|||
-/
|
||||
prelude
|
||||
import Init.Data.List.Impl
|
||||
import Init.Data.List.Nat.TakeDrop
|
||||
|
||||
/-!
|
||||
# Definition of `merge` and `mergeSort`.
|
||||
|
|
|
|||
|
|
@ -6,6 +6,7 @@ Authors: Kim Morrison
|
|||
prelude
|
||||
import Init.Data.List.Perm
|
||||
import Init.Data.List.Sort.Basic
|
||||
import Init.Data.List.Nat.Range
|
||||
import Init.Data.Bool
|
||||
|
||||
/-!
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue