File: Mathlib/Algebra/Order/Ring/Cast.lean

full_name: Int.cast_mono
code: lemma cast_mono : Monotone (Int.cast : ℤ → R) 
goal: Monotone Int.cast

full_name: Int.cast_nonneg
code: @[simp] lemma cast_nonneg : ∀ {n : ℤ}, (0 : R) ≤ n ↔ 0 ≤ n
  
goal: 0 ≤ ↑n ↔ 0 ≤ n

full_name: Int.cast_le
code: @[simp, norm_cast] lemma cast_le : (m : R) ≤ n ↔ m ≤ n 
goal: ↑m ≤ ↑n ↔ m ≤ n

full_name: Int.cast_strictMono
code: lemma cast_strictMono : StrictMono (fun x : ℤ => (x : R)) 
goal: StrictMono fun x => ↑x

full_name: Int.cast_lt
code: @[simp, norm_cast] lemma cast_lt : (m : R) < n ↔ m < n 
goal: ↑m < ↑n ↔ m < n

full_name: Int.cast_nonpos
code: @[simp] lemma cast_nonpos : (n : R) ≤ 0 ↔ n ≤ 0 
goal: ↑n ≤ 0 ↔ n ≤ 0

full_name: Int.cast_pos
code: @[simp] lemma cast_pos : (0 : R) < n ↔ 0 < n 
goal: 0 < ↑n ↔ 0 < n

full_name: Int.cast_lt_zero
code: @[simp] lemma cast_lt_zero : (n : R) < 0 ↔ n < 0 
goal: ↑n < 0 ↔ n < 0

full_name: Int.cast_min
code: @[simp, norm_cast]
lemma cast_min : ↑(min a b) = (min a b : R) 
goal: ↑(min a b) = min ↑a ↑b

full_name: Int.cast_max
code: @[simp, norm_cast]
lemma cast_max : (↑(max a b) : R) = max (a : R) (b : R) 
goal: ↑(max a b) = max ↑a ↑b

full_name: Int.cast_abs
code: @[simp, norm_cast]
lemma cast_abs : (↑|a| : R) = |(a : R)| 
goal: ↑|a| = |↑a|

full_name: Int.cast_one_le_of_pos
code: lemma cast_one_le_of_pos (h : 0 < a) : (1 : R) ≤ a 
goal: 1 ≤ ↑a

full_name: Int.cast_le_neg_one_of_neg
code: lemma cast_le_neg_one_of_neg (h : a < 0) : (a : R) ≤ -1 
goal: ↑a ≤ -1

full_name: Int.cast_le_neg_one_or_one_le_cast_of_ne_zero
code: lemma cast_le_neg_one_or_one_le_cast_of_ne_zero (hn : n ≠ 0) : (n : R) ≤ -1 ∨ 1 ≤ (n : R) 
goal: ↑n ≤ -1 ∨ 1 ≤ ↑n

full_name: Int.nneg_mul_add_sq_of_abs_le_one
code: lemma nneg_mul_add_sq_of_abs_le_one (n : ℤ) (hx : |x| ≤ 1) : (0 : R) ≤ n * x + n * n 
goal: 0 ≤ ↑n * x + ↑n * ↑n

full_name: Int.cast_natAbs
code: lemma cast_natAbs : (n.natAbs : R) = |n| 
goal: ↑n.natAbs = ↑|n|

full_name: OrderDual.instIntCast
code: instance instIntCast             [IntCast R]             : IntCast Rᵒᵈ             
goal: IntCast Rᵒᵈ

full_name: OrderDual.instAddGroupWithOne
code: instance instAddGroupWithOne     [AddGroupWithOne R]     : AddGroupWithOne Rᵒᵈ     
goal: AddGroupWithOne Rᵒᵈ

full_name: OrderDual.instAddCommGroupWithOne
code: instance instAddCommGroupWithOne [AddCommGroupWithOne R] : AddCommGroupWithOne Rᵒᵈ 
goal: AddCommGroupWithOne Rᵒᵈ

full_name: toDual_intCast
code: @[simp] lemma toDual_intCast [IntCast R] (n : ℤ) : toDual (n : R) = n 
goal: OrderDual.toDual ↑n = ↑n

full_name: ofDual_intCast
code: @[simp] lemma ofDual_intCast [IntCast R] (n : ℤ) : (ofDual n : R) = n 
goal: OrderDual.ofDual ↑n = ↑n

full_name: Lex.instIntCast
code: instance instIntCast             [IntCast R]             : IntCast (Lex R)             
goal: IntCast (Lex R)

full_name: Lex.instAddGroupWithOne
code: instance instAddGroupWithOne     [AddGroupWithOne R]     : AddGroupWithOne (Lex R)     
goal: AddGroupWithOne (Lex R)

full_name: Lex.instAddCommGroupWithOne
code: instance instAddCommGroupWithOne [AddCommGroupWithOne R] : AddCommGroupWithOne (Lex R) 
goal: AddCommGroupWithOne (Lex R)

full_name: toLex_intCast
code: @[simp] lemma toLex_intCast [IntCast R] (n : ℤ) : toLex (n : R) = n 
goal: toLex ↑n = ↑n

full_name: ofLex_intCast
code: @[simp] lemma ofLex_intCast [IntCast R] (n : ℤ) : (ofLex n : R) = n 
goal: ofLex ↑n = ↑n