File: Mathlib/Data/Rat/Cast/Order.lean

full_name: Rat.castHom_rat
code: @[simp]
theorem castHom_rat : castHom ℚ = RingHom.id ℚ 
goal: Rat.castHom ℚ = RingHom.id ℚ

full_name: Rat.cast_pos_of_pos
code: theorem cast_pos_of_pos (hq : 0 < q) : (0 : K) < q 
goal: 0 < ↑q

full_name: Rat.cast_strictMono
code: @[mono]
theorem cast_strictMono : StrictMono ((↑) : ℚ → K) 
goal: StrictMono Rat.cast

full_name: Rat.cast_mono
code: @[mono]
theorem cast_mono : Monotone ((↑) : ℚ → K) 
goal: Monotone Rat.cast

full_name: Rat.castOrderEmbedding_apply
code: /-- Coercion from `ℚ` as an order embedding. -/
@[simps!]
def castOrderEmbedding : ℚ ↪o K 
goal: ∀ (a : ℚ), Rat.castOrderEmbedding a = ↑a

full_name: Rat.castOrderEmbedding
code: /-- Coercion from `ℚ` as an order embedding. -/
@[simps!]
def castOrderEmbedding : ℚ ↪o K 
goal: ℚ ↪o K

full_name: Rat.cast_le
code: @[simp, norm_cast] lemma cast_le : (p : K) ≤ q ↔ p ≤ q 
goal: ↑p ≤ ↑q ↔ p ≤ q

full_name: Rat.cast_lt
code: @[simp, norm_cast] lemma cast_lt : (p : K) < q ↔ p < q 
goal: ↑p < ↑q ↔ p < q

full_name: Rat.cast_nonneg
code: @[simp] lemma cast_nonneg : 0 ≤ (q : K) ↔ 0 ≤ q 
goal: 0 ≤ ↑q ↔ 0 ≤ q

full_name: Rat.cast_nonpos
code: @[simp] lemma cast_nonpos : (q : K) ≤ 0 ↔ q ≤ 0 
goal: ↑q ≤ 0 ↔ q ≤ 0

full_name: Rat.cast_pos
code: @[simp] lemma cast_pos : (0 : K) < q ↔ 0 < q 
goal: 0 < ↑q ↔ 0 < q

full_name: Rat.cast_lt_zero
code: @[simp] lemma cast_lt_zero : (q : K) < 0 ↔ q < 0 
goal: ↑q < 0 ↔ q < 0

full_name: Rat.cast_min
code: @[simp, norm_cast]
lemma cast_min (p q : ℚ) : (↑(min p q) : K) = min (p : K) (q : K) 
goal: ↑(min p q) = min ↑p ↑q

full_name: Rat.cast_max
code: @[simp, norm_cast]
lemma cast_max (p q : ℚ) : (↑(max p q) : K) = max (p : K) (q : K) 
goal: ↑(max p q) = max ↑p ↑q

full_name: Rat.cast_abs
code: @[simp, norm_cast] lemma cast_abs (q : ℚ) : ((|q| : ℚ) : K) = |(q : K)| 
goal: ↑|q| = |↑q|

full_name: Rat.preimage_cast_Icc
code: @[simp]
theorem preimage_cast_Icc (p q : ℚ) : (↑) ⁻¹' Icc (p : K) q = Icc p q 
goal: Rat.cast ⁻¹' Set.Icc ↑p ↑q = Set.Icc p q

full_name: Rat.preimage_cast_Ico
code: @[simp]
theorem preimage_cast_Ico (p q : ℚ) : (↑) ⁻¹' Ico (p : K) q = Ico p q 
goal: Rat.cast ⁻¹' Set.Ico ↑p ↑q = Set.Ico p q

full_name: Rat.preimage_cast_Ioc
code: @[simp]
theorem preimage_cast_Ioc (p q : ℚ) : (↑) ⁻¹' Ioc (p : K) q = Ioc p q 
goal: Rat.cast ⁻¹' Set.Ioc ↑p ↑q = Set.Ioc p q

full_name: Rat.preimage_cast_Ioo
code: @[simp]
theorem preimage_cast_Ioo (p q : ℚ) : (↑) ⁻¹' Ioo (p : K) q = Ioo p q 
goal: Rat.cast ⁻¹' Set.Ioo ↑p ↑q = Set.Ioo p q

full_name: Rat.preimage_cast_Ici
code: @[simp]
theorem preimage_cast_Ici (q : ℚ) : (↑) ⁻¹' Ici (q : K) = Ici q 
goal: Rat.cast ⁻¹' Set.Ici ↑q = Set.Ici q

full_name: Rat.preimage_cast_Iic
code: @[simp]
theorem preimage_cast_Iic (q : ℚ) : (↑) ⁻¹' Iic (q : K) = Iic q 
goal: Rat.cast ⁻¹' Set.Iic ↑q = Set.Iic q

full_name: Rat.preimage_cast_Ioi
code: @[simp]
theorem preimage_cast_Ioi (q : ℚ) : (↑) ⁻¹' Ioi (q : K) = Ioi q 
goal: Rat.cast ⁻¹' Set.Ioi ↑q = Set.Ioi q

full_name: Rat.preimage_cast_Iio
code: @[simp]
theorem preimage_cast_Iio (q : ℚ) : (↑) ⁻¹' Iio (q : K) = Iio q 
goal: Rat.cast ⁻¹' Set.Iio ↑q = Set.Iio q

full_name: Rat.preimage_cast_uIcc
code: @[simp]
theorem preimage_cast_uIcc (p q : ℚ) : (↑) ⁻¹' uIcc (p : K) q = uIcc p q 
goal: Rat.cast ⁻¹' Set.uIcc ↑p ↑q = Set.uIcc p q

full_name: Rat.preimage_cast_uIoc
code: @[simp]
theorem preimage_cast_uIoc (p q : ℚ) : (↑) ⁻¹' uIoc (p : K) q = uIoc p q 
goal: Rat.cast ⁻¹' Ι ↑p ↑q = Ι p q

full_name: NNRat.cast_strictMono
code: theorem cast_strictMono : StrictMono ((↑) : ℚ≥0 → K) 
goal: StrictMono NNRat.cast

full_name: NNRat.cast_mono
code: @[mono]
theorem cast_mono : Monotone ((↑) : ℚ≥0 → K) 
goal: Monotone NNRat.cast

full_name: NNRat.castOrderEmbedding_apply
code: /-- Coercion from `ℚ` as an order embedding. -/
@[simps!]
def castOrderEmbedding : ℚ≥0 ↪o K 
goal: ∀ (a : ℚ≥0), NNRat.castOrderEmbedding a = ↑a

full_name: NNRat.castOrderEmbedding
code: /-- Coercion from `ℚ` as an order embedding. -/
@[simps!]
def castOrderEmbedding : ℚ≥0 ↪o K 
goal: ℚ≥0 ↪o K

full_name: NNRat.cast_le
code: @[simp, norm_cast] lemma cast_le : (p : K) ≤ q ↔ p ≤ q 
goal: ↑p ≤ ↑q ↔ p ≤ q

full_name: NNRat.cast_lt
code: @[simp, norm_cast] lemma cast_lt : (p : K) < q ↔ p < q 
goal: ↑p < ↑q ↔ p < q

full_name: NNRat.cast_nonpos
code: @[simp] lemma cast_nonpos : (q : K) ≤ 0 ↔ q ≤ 0 
goal: ↑q ≤ 0 ↔ q ≤ 0

full_name: NNRat.cast_pos
code: @[simp] lemma cast_pos : (0 : K) < q ↔ 0 < q 
goal: 0 < ↑q ↔ 0 < q

full_name: NNRat.cast_lt_zero
code: @[norm_cast] lemma cast_lt_zero : (q : K) < 0 ↔ q < 0 
goal: ↑q < 0 ↔ q < 0

full_name: NNRat.not_cast_lt_zero
code: @[simp] lemma not_cast_lt_zero : ¬(q : K) < 0 
goal: ¬↑q < 0

full_name: NNRat.cast_min
code: @[simp, norm_cast] lemma cast_min (p q : ℚ≥0) : (↑(min p q) : K) = min (p : K) (q : K) 
goal: ↑(min p q) = min ↑p ↑q

full_name: NNRat.cast_max
code: @[simp, norm_cast] lemma cast_max (p q : ℚ≥0) : (↑(max p q) : K) = max (p : K) (q : K) 
goal: ↑(max p q) = max ↑p ↑q

full_name: NNRat.preimage_cast_Icc
code: @[simp]
theorem preimage_cast_Icc (p q : ℚ≥0) : (↑) ⁻¹' Icc (p : K) q = Icc p q 
goal: NNRat.cast ⁻¹' Set.Icc ↑p ↑q = Set.Icc p q

full_name: NNRat.preimage_cast_Ico
code: @[simp]
theorem preimage_cast_Ico (p q : ℚ≥0) : (↑) ⁻¹' Ico (p : K) q = Ico p q 
goal: NNRat.cast ⁻¹' Set.Ico ↑p ↑q = Set.Ico p q

full_name: NNRat.preimage_cast_Ioc
code: @[simp]
theorem preimage_cast_Ioc (p q : ℚ≥0) : (↑) ⁻¹' Ioc (p : K) q = Ioc p q 
goal: NNRat.cast ⁻¹' Set.Ioc ↑p ↑q = Set.Ioc p q

full_name: NNRat.preimage_cast_Ioo
code: @[simp]
theorem preimage_cast_Ioo (p q : ℚ≥0) : (↑) ⁻¹' Ioo (p : K) q = Ioo p q 
goal: NNRat.cast ⁻¹' Set.Ioo ↑p ↑q = Set.Ioo p q

full_name: NNRat.preimage_cast_Ici
code: @[simp]
theorem preimage_cast_Ici (p : ℚ≥0) : (↑) ⁻¹' Ici (p : K) = Ici p 
goal: NNRat.cast ⁻¹' Set.Ici ↑p = Set.Ici p

full_name: NNRat.preimage_cast_Iic
code: @[simp]
theorem preimage_cast_Iic (p : ℚ≥0) : (↑) ⁻¹' Iic (p : K) = Iic p 
goal: NNRat.cast ⁻¹' Set.Iic ↑p = Set.Iic p

full_name: NNRat.preimage_cast_Ioi
code: @[simp]
theorem preimage_cast_Ioi (p : ℚ≥0) : (↑) ⁻¹' Ioi (p : K) = Ioi p 
goal: NNRat.cast ⁻¹' Set.Ioi ↑p = Set.Ioi p

full_name: NNRat.preimage_cast_Iio
code: @[simp]
theorem preimage_cast_Iio (p : ℚ≥0) : (↑) ⁻¹' Iio (p : K) = Iio p 
goal: NNRat.cast ⁻¹' Set.Iio ↑p = Set.Iio p

full_name: NNRat.preimage_cast_uIcc
code: @[simp]
theorem preimage_cast_uIcc (p q : ℚ≥0) : (↑) ⁻¹' uIcc (p : K) q = uIcc p q 
goal: NNRat.cast ⁻¹' Set.uIcc ↑p ↑q = Set.uIcc p q

full_name: NNRat.preimage_cast_uIoc
code: @[simp]
theorem preimage_cast_uIoc (p q : ℚ≥0) : (↑) ⁻¹' uIoc (p : K) q = uIoc p q 
goal: NNRat.cast ⁻¹' Ι ↑p ↑q = Ι p q

full_name: Mathlib.Meta.Positivity.evalRatCast
code: /-- Extension for Rat.cast. -/
@[positivity Rat.cast _]
def evalRatCast : PositivityExt 
goal: Mathlib.Meta.Positivity.PositivityExt

full_name: Mathlib.Meta.Positivity.evalNNRatCast
code: /-- Extension for NNRat.cast. -/
@[positivity NNRat.cast _]
def evalNNRatCast : PositivityExt 
goal: Mathlib.Meta.Positivity.PositivityExt