state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
a b c : ℚ
this : Rat.Nonneg (b - a) ∨ Rat.Nonneg (-(b - a))
⊢ Rat.Nonneg (b - a) ∨ Rat.Nonneg (a - b) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rwa [neg_sub] at this | protected theorem le_total : a ≤ b ∨ b ≤ a := by
have := Rat.nonneg_total (b - a)
rw [Rat.le_iff_Nonneg, Rat.le_iff_Nonneg]
| Mathlib.Data.Rat.Order.166_0.NTjR6KCugNscheB | protected theorem le_total : a ≤ b ∨ b ≤ a | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
hab : a ≤ b
hba : b ≤ a
⊢ a = b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Rat.le_iff_Nonneg] at hab hba | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b := by
| Mathlib.Data.Rat.Order.172_0.NTjR6KCugNscheB | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
hab : Rat.Nonneg (b - a)
hba : Rat.Nonneg (a - b)
⊢ a = b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [sub_eq_add_neg] at hba | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b := by
rw [Rat.le_iff_Nonneg] at hab hba
| Mathlib.Data.Rat.Order.172_0.NTjR6KCugNscheB | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
hab : Rat.Nonneg (b - a)
hba : Rat.Nonneg (a + -b)
⊢ a = b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← neg_sub, sub_eq_add_neg] at hab | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b := by
rw [Rat.le_iff_Nonneg] at hab hba
rw [sub_eq_add_neg] at hba
| Mathlib.Data.Rat.Order.172_0.NTjR6KCugNscheB | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
hab : Rat.Nonneg (-(a + -b))
hba : Rat.Nonneg (a + -b)
⊢ a = b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | have := eq_neg_of_add_eq_zero_left (Rat.nonneg_antisymm hba hab) | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b := by
rw [Rat.le_iff_Nonneg] at hab hba
rw [sub_eq_add_neg] at hba
rw [← neg_sub, sub_eq_add_neg] at hab
| Mathlib.Data.Rat.Order.172_0.NTjR6KCugNscheB | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
hab : Rat.Nonneg (-(a + -b))
hba : Rat.Nonneg (a + -b)
this : a = - -b
⊢ a = b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rwa [neg_neg] at this | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b := by
rw [Rat.le_iff_Nonneg] at hab hba
rw [sub_eq_add_neg] at hba
rw [← neg_sub, sub_eq_add_neg] at hab
have := eq_neg_of_add_eq_zero_left (Rat.nonneg_antisymm hba hab)
| Mathlib.Data.Rat.Order.172_0.NTjR6KCugNscheB | protected theorem le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) : a = b | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
hab : a ≤ b
hbc : b ≤ c
⊢ a ≤ c | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Rat.le_iff_Nonneg] at hab hbc | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c := by
| Mathlib.Data.Rat.Order.180_0.NTjR6KCugNscheB | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
hab : Rat.Nonneg (b - a)
hbc : Rat.Nonneg (c - b)
⊢ a ≤ c | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | have : Rat.Nonneg (b - a + (c - b)) := Rat.nonneg_add hab hbc | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c := by
rw [Rat.le_iff_Nonneg] at hab hbc
| Mathlib.Data.Rat.Order.180_0.NTjR6KCugNscheB | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
hab : Rat.Nonneg (b - a)
hbc : Rat.Nonneg (c - b)
this : Rat.Nonneg (b - a + (c - b))
⊢ a ≤ c | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp_rw [sub_eq_add_neg, add_left_comm (b + -a) c (-b), add_comm (b + -a) (-b),
add_left_comm (-b) b (-a), add_comm (-b) (-a), add_neg_cancel_comm_assoc,
← sub_eq_add_neg] at this | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c := by
rw [Rat.le_iff_Nonneg] at hab hbc
have : Rat.Nonneg (b - a + (c - b)) := Rat.nonneg_add hab hbc
| Mathlib.Data.Rat.Order.180_0.NTjR6KCugNscheB | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
hab : Rat.Nonneg (b - a)
hbc : Rat.Nonneg (c - b)
this : Rat.Nonneg (c - a)
⊢ a ≤ c | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Rat.le_iff_Nonneg] | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c := by
rw [Rat.le_iff_Nonneg] at hab hbc
have : Rat.Nonneg (b - a + (c - b)) := Rat.nonneg_add hab hbc
simp_rw [sub_eq_add_neg, add_left_comm (b + -a) c (-b), add_comm (b + -a) (-b),
add_left_comm (-b) b (-a), add_comm (-b) (-a), add_ne... | Mathlib.Data.Rat.Order.180_0.NTjR6KCugNscheB | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
hab : Rat.Nonneg (b - a)
hbc : Rat.Nonneg (c - b)
this : Rat.Nonneg (c - a)
⊢ Rat.Nonneg (c - a) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact this | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c := by
rw [Rat.le_iff_Nonneg] at hab hbc
have : Rat.Nonneg (b - a + (c - b)) := Rat.nonneg_add hab hbc
simp_rw [sub_eq_add_neg, add_left_comm (b + -a) c (-b), add_comm (b + -a) (-b),
add_left_comm (-b) b (-a), add_comm (-b) (-a), add_ne... | Mathlib.Data.Rat.Order.180_0.NTjR6KCugNscheB | protected theorem le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c | Mathlib_Data_Rat_Order |
a b c x✝¹ x✝ : ℚ
⊢ x✝¹ < x✝ ↔ x✝¹ ≤ x✝ ∧ ¬x✝ ≤ x✝¹ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← Rat.not_le, and_iff_right_of_imp (Rat.le_total _ _).resolve_left] | instance linearOrder : LinearOrder ℚ where
le_refl := Rat.le_refl
le_trans := @Rat.le_trans
le_antisymm := @Rat.le_antisymm
le_total := Rat.le_total
decidableLE _ _ := by infer_instance
lt_iff_le_not_le _ _ := by
| Mathlib.Data.Rat.Order.192_0.NTjR6KCugNscheB | instance linearOrder : LinearOrder ℚ where
le_refl | Mathlib_Data_Rat_Order |
a b c x✝¹ x✝ : ℚ
⊢ Decidable ((fun x x_1 => x ≤ x_1) x✝¹ x✝) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance linearOrder : LinearOrder ℚ where
le_refl := Rat.le_refl
le_trans := @Rat.le_trans
le_antisymm := @Rat.le_antisymm
le_total := Rat.le_total
decidableLE _ _ := by | Mathlib.Data.Rat.Order.192_0.NTjR6KCugNscheB | instance linearOrder : LinearOrder ℚ where
le_refl | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ LT ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : LT ℚ := by | Mathlib.Data.Rat.Order.202_0.NTjR6KCugNscheB | instance : LT ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ DistribLattice ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : DistribLattice ℚ := by | Mathlib.Data.Rat.Order.204_0.NTjR6KCugNscheB | instance : DistribLattice ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ Lattice ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : Lattice ℚ := by | Mathlib.Data.Rat.Order.206_0.NTjR6KCugNscheB | instance : Lattice ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ SemilatticeInf ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : SemilatticeInf ℚ := by | Mathlib.Data.Rat.Order.208_0.NTjR6KCugNscheB | instance : SemilatticeInf ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ SemilatticeSup ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : SemilatticeSup ℚ := by | Mathlib.Data.Rat.Order.210_0.NTjR6KCugNscheB | instance : SemilatticeSup ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ Inf ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : Inf ℚ := by | Mathlib.Data.Rat.Order.212_0.NTjR6KCugNscheB | instance : Inf ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ Sup ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : Sup ℚ := by | Mathlib.Data.Rat.Order.214_0.NTjR6KCugNscheB | instance : Sup ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ PartialOrder ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : PartialOrder ℚ := by | Mathlib.Data.Rat.Order.216_0.NTjR6KCugNscheB | instance : PartialOrder ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ Preorder ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : Preorder ℚ := by | Mathlib.Data.Rat.Order.218_0.NTjR6KCugNscheB | instance : Preorder ℚ | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p ≤ q ↔ p.num * ↑q.den ≤ q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← @num_den q, ← @num_den p] | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
| Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p.num /. ↑p.den ≤ q.num /. ↑q.den ↔
(p.num /. ↑p.den).num * ↑(q.num /. ↑q.den).den ≤ (q.num /. ↑q.den).num * ↑(p.num /. ↑p.den).den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | conv_rhs => simp only [num_den] | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
rw [← @num_den q, ← @num_den p]
| Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
| (p.num /. ↑p.den).num * ↑(q.num /. ↑q.den).den ≤ (q.num /. ↑q.den).num * ↑(p.num /. ↑p.den).den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp only [num_den] | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
rw [← @num_den q, ← @num_den p]
conv_rhs => | Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
| (p.num /. ↑p.den).num * ↑(q.num /. ↑q.den).den ≤ (q.num /. ↑q.den).num * ↑(p.num /. ↑p.den).den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp only [num_den] | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
rw [← @num_den q, ← @num_den p]
conv_rhs => | Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
| (p.num /. ↑p.den).num * ↑(q.num /. ↑q.den).den ≤ (q.num /. ↑q.den).num * ↑(p.num /. ↑p.den).den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp only [num_den] | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
rw [← @num_den q, ← @num_den p]
conv_rhs => | Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p.num /. ↑p.den ≤ q.num /. ↑q.den ↔ p.num * ↑q.den ≤ q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact Rat.le_def (mod_cast p.pos) (mod_cast q.pos) | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den := by
rw [← @num_den q, ← @num_den p]
conv_rhs => simp only [num_den]
| Mathlib.Data.Rat.Order.220_0.NTjR6KCugNscheB | protected theorem le_def' {p q : ℚ} : p ≤ q ↔ p.num * q.den ≤ q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p < q ↔ p.num * ↑q.den < q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [lt_iff_le_and_ne, Rat.le_def'] | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
| Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q ↔ p.num * ↑q.den < q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> intro h
· exact lt_iff_le_and_ne.mpr ⟨h.left, this.mp h.right⟩
· have tmp := lt_iff_le_and_ne.mp h
exact ⟨tmp.left, this.mpr tmp.right⟩ | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
| Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
⊢ p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q ↔ p.num * ↑q.den < q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | constructor | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
| Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
case mp
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
⊢ p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q → p.num * ↑q.den < q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | intro h | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> | Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
case mpr
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
⊢ p.num * ↑q.den < q.num * ↑p.den → p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | intro h | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> | Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
case mp
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
h : p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q
⊢ p.num * ↑q.den < q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact lt_iff_le_and_ne.mpr ⟨h.left, this.mp h.right⟩ | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> intro h
· | Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
case mpr
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
h : p.num * ↑q.den < q.num * ↑p.den
⊢ p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | have tmp := lt_iff_le_and_ne.mp h | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> intro h
· exact lt_iff_le_and_ne.mpr ⟨h.left, this.mp h.right⟩
· | Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
case mpr
a b c p q : ℚ
this : p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den
h : p.num * ↑q.den < q.num * ↑p.den
tmp : p.num * ↑q.den ≤ q.num * ↑p.den ∧ p.num * ↑q.den ≠ q.num * ↑p.den
⊢ p.num * ↑q.den ≤ q.num * ↑p.den ∧ p ≠ q | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact ⟨tmp.left, this.mpr tmp.right⟩ | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> intro h
· exact lt_iff_le_and_ne.mpr ⟨h.left, this.mp h.right⟩
· have tmp := lt_iff_le_and_ne.mp h
| Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
a b c p q : ℚ
⊢ p ≠ q ↔ p.num * ↑q.den ≠ q.num * ↑p.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact not_iff_not.mpr eq_iff_mul_eq_mul | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den := by
rw [lt_iff_le_and_ne, Rat.le_def']
suffices p ≠ q ↔ p.num * q.den ≠ q.num * p.den by
constructor <;> intro h
· exact lt_iff_le_and_ne.mpr ⟨h.left, this.mp h.right⟩
· have tmp := lt_iff_le_and_ne.mp h
exact ⟨tmp.left, ... | Mathlib.Data.Rat.Order.226_0.NTjR6KCugNscheB | protected theorem lt_def {p q : ℚ} : p < q ↔ p.num * q.den < q.num * p.den | Mathlib_Data_Rat_Order |
a✝ b c a : ℚ
⊢ Rat.Nonneg a ↔ 0 ≤ a | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Rat.le_iff_Nonneg] | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a := by
| Mathlib.Data.Rat.Order.236_0.NTjR6KCugNscheB | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a | Mathlib_Data_Rat_Order |
a✝ b c a : ℚ
⊢ Rat.Nonneg a ↔ Rat.Nonneg (a - 0) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | show Rat.Nonneg a ↔ Rat.Nonneg (a - 0) | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a := by
rw [Rat.le_iff_Nonneg]
| Mathlib.Data.Rat.Order.236_0.NTjR6KCugNscheB | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a | Mathlib_Data_Rat_Order |
a✝ b c a : ℚ
⊢ Rat.Nonneg a ↔ Rat.Nonneg (a - 0) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a := by
rw [Rat.le_iff_Nonneg]
show Rat.Nonneg a ↔ Rat.Nonneg (a - 0)
| Mathlib.Data.Rat.Order.236_0.NTjR6KCugNscheB | theorem nonneg_iff_zero_le {a} : Rat.Nonneg a ↔ 0 ≤ a | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ a b c : ℚ
⊢ c + a ≤ c + b ↔ a ≤ b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Rat.le_iff_Nonneg, add_sub_add_left_eq_sub, Rat.le_iff_Nonneg] | protected theorem add_le_add_left {a b c : ℚ} : c + a ≤ c + b ↔ a ≤ b := by
| Mathlib.Data.Rat.Order.246_0.NTjR6KCugNscheB | protected theorem add_le_add_left {a b c : ℚ} : c + a ≤ c + b ↔ a ≤ b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0 ≤ a * b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← nonneg_iff_zero_le] at ha hb ⊢ | protected theorem mul_nonneg {a b : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b := by
| Mathlib.Data.Rat.Order.250_0.NTjR6KCugNscheB | protected theorem mul_nonneg {a b : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b | Mathlib_Data_Rat_Order |
a✝ b✝ c a b : ℚ
ha : Rat.Nonneg a
hb : Rat.Nonneg b
⊢ Rat.Nonneg (a * b) | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | exact Rat.nonneg_mul ha hb | protected theorem mul_nonneg {a b : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b := by
rw [← nonneg_iff_zero_le] at ha hb ⊢; | Mathlib.Data.Rat.Order.250_0.NTjR6KCugNscheB | protected theorem mul_nonneg {a b : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b | Mathlib_Data_Rat_Order |
a b c : ℚ
src✝² : Field ℚ := field
src✝¹ : LinearOrder ℚ := linearOrder
src✝ : Semiring ℚ := semiring
⊢ 0 ≤ 1 | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | decide | instance : LinearOrderedField ℚ :=
{ Rat.field, Rat.linearOrder, Rat.semiring with
zero_le_one := by | Mathlib.Data.Rat.Order.254_0.NTjR6KCugNscheB | instance : LinearOrderedField ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ LinearOrderedCommRing ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : LinearOrderedCommRing ℚ := by | Mathlib.Data.Rat.Order.263_0.NTjR6KCugNscheB | instance : LinearOrderedCommRing ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ LinearOrderedRing ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : LinearOrderedRing ℚ := by | Mathlib.Data.Rat.Order.265_0.NTjR6KCugNscheB | instance : LinearOrderedRing ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ OrderedRing ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : OrderedRing ℚ := by | Mathlib.Data.Rat.Order.267_0.NTjR6KCugNscheB | instance : OrderedRing ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ LinearOrderedSemiring ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : LinearOrderedSemiring ℚ := by | Mathlib.Data.Rat.Order.269_0.NTjR6KCugNscheB | instance : LinearOrderedSemiring ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ OrderedSemiring ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : OrderedSemiring ℚ := by | Mathlib.Data.Rat.Order.271_0.NTjR6KCugNscheB | instance : OrderedSemiring ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ LinearOrderedAddCommGroup ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : LinearOrderedAddCommGroup ℚ := by | Mathlib.Data.Rat.Order.273_0.NTjR6KCugNscheB | instance : LinearOrderedAddCommGroup ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ OrderedAddCommGroup ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : OrderedAddCommGroup ℚ := by | Mathlib.Data.Rat.Order.275_0.NTjR6KCugNscheB | instance : OrderedAddCommGroup ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ OrderedCancelAddCommMonoid ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : OrderedCancelAddCommMonoid ℚ := by | Mathlib.Data.Rat.Order.277_0.NTjR6KCugNscheB | instance : OrderedCancelAddCommMonoid ℚ | Mathlib_Data_Rat_Order |
a b c : ℚ
⊢ OrderedAddCommMonoid ℚ | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | infer_instance | instance : OrderedAddCommMonoid ℚ := by | Mathlib.Data.Rat.Order.279_0.NTjR6KCugNscheB | instance : OrderedAddCommMonoid ℚ | Mathlib_Data_Rat_Order |
a✝ b c a : ℚ
⊢ a.num ≤ 0 ↔ a ≤ 0 | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simpa [(by cases a; rfl : (-a).num = -a.num)] using @num_nonneg_iff_zero_le (-a) | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a :=
lt_iff_lt_of_le_iff_le <| by
| Mathlib.Data.Rat.Order.281_0.NTjR6KCugNscheB | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a | Mathlib_Data_Rat_Order |
a✝ b c a : ℚ
⊢ (-a).num = -a.num | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | cases a | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a :=
lt_iff_lt_of_le_iff_le <| by
simpa [(by | Mathlib.Data.Rat.Order.281_0.NTjR6KCugNscheB | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a | Mathlib_Data_Rat_Order |
case mk'
a b c : ℚ
num✝ : ℤ
den✝ : ℕ
den_nz✝ : den✝ ≠ 0
reduced✝ : Nat.Coprime (Int.natAbs num✝) den✝
⊢ (-mk' num✝ den✝).num = -(mk' num✝ den✝).num | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rfl | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a :=
lt_iff_lt_of_le_iff_le <| by
simpa [(by cases a; | Mathlib.Data.Rat.Order.281_0.NTjR6KCugNscheB | theorem num_pos_iff_pos {a : ℚ} : 0 < a.num ↔ 0 < a | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ : ℚ
a b c d : ℤ
b_pos : 0 < b
d_pos : 0 < d
⊢ ↑a / ↑b < ↑c / ↑d ↔ a * d < c * b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp only [lt_iff_le_not_le] | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by
| Mathlib.Data.Rat.Order.286_0.NTjR6KCugNscheB | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b | Mathlib_Data_Rat_Order |
a✝ b✝ c✝ : ℚ
a b c d : ℤ
b_pos : 0 < b
d_pos : 0 < d
⊢ ↑a / ↑b ≤ ↑c / ↑d ∧ ¬↑c / ↑d ≤ ↑a / ↑b ↔ a * d ≤ c * b ∧ ¬c * b ≤ a * d | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | apply and_congr | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by
simp only [lt_iff_le_not_le]
| Mathlib.Data.Rat.Order.286_0.NTjR6KCugNscheB | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b | Mathlib_Data_Rat_Order |
case h₁
a✝ b✝ c✝ : ℚ
a b c d : ℤ
b_pos : 0 < b
d_pos : 0 < d
⊢ ↑a / ↑b ≤ ↑c / ↑d ↔ a * d ≤ c * b | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp [div_num_den, Rat.le_def b_pos d_pos] | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by
simp only [lt_iff_le_not_le]
apply and_congr
· | Mathlib.Data.Rat.Order.286_0.NTjR6KCugNscheB | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b | Mathlib_Data_Rat_Order |
case h₂
a✝ b✝ c✝ : ℚ
a b c d : ℤ
b_pos : 0 < b
d_pos : 0 < d
⊢ ¬↑c / ↑d ≤ ↑a / ↑b ↔ ¬c * b ≤ a * d | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | apply not_congr | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by
simp only [lt_iff_le_not_le]
apply and_congr
· simp [div_num_den, Rat.le_def b_pos d_pos]
· | Mathlib.Data.Rat.Order.286_0.NTjR6KCugNscheB | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b | Mathlib_Data_Rat_Order |
case h₂.h
a✝ b✝ c✝ : ℚ
a b c d : ℤ
b_pos : 0 < b
d_pos : 0 < d
⊢ ↑c / ↑d ≤ ↑a / ↑b ↔ c * b ≤ a * d | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp [div_num_den, Rat.le_def d_pos b_pos] | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by
simp only [lt_iff_le_not_le]
apply and_congr
· simp [div_num_den, Rat.le_def b_pos d_pos]
· apply not_congr
| Mathlib.Data.Rat.Order.286_0.NTjR6KCugNscheB | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b | Mathlib_Data_Rat_Order |
a b c q : ℚ
⊢ q < 1 ↔ q.num < ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | simp [Rat.lt_def] | theorem lt_one_iff_num_lt_denom {q : ℚ} : q < 1 ↔ q.num < q.den := by | Mathlib.Data.Rat.Order.295_0.NTjR6KCugNscheB | theorem lt_one_iff_num_lt_denom {q : ℚ} : q < 1 ↔ q.num < q.den | Mathlib_Data_Rat_Order |
a b c q : ℚ
⊢ |q| = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rcases le_total q 0 with hq | hq | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
| Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inl
a b c q : ℚ
hq : q ≤ 0
⊢ |q| = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [abs_of_nonpos hq] | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· | Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inl
a b c q : ℚ
hq : q ≤ 0
⊢ -q = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← @num_den q, ← divInt_zero_one, Rat.le_def (Int.coe_nat_pos.2 q.pos) zero_lt_one, mul_one,
zero_mul] at hq | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· rw [abs_of_nonpos hq]
| Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inl
a b c q : ℚ
hq : q.num ≤ 0
⊢ -q = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Int.ofNat_natAbs_of_nonpos hq, ← neg_def, num_den] | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· rw [abs_of_nonpos hq]
rw [← @num_den q, ← divInt_zero_one, Rat.le_def (Int.coe_nat_pos.2 q.pos) zero_lt_one, mul_one,
zero_mul] at hq
| Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inr
a b c q : ℚ
hq : 0 ≤ q
⊢ |q| = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [abs_of_nonneg hq] | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· rw [abs_of_nonpos hq]
rw [← @num_den q, ← divInt_zero_one, Rat.le_def (Int.coe_nat_pos.2 q.pos) zero_lt_one, mul_one,
zero_mul] at hq
rw [Int.ofNat_natAbs_of_nonpos hq, ← neg_def, num_den]
· | Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inr
a b c q : ℚ
hq : 0 ≤ q
⊢ q = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [← @num_den q, ← divInt_zero_one, Rat.le_def zero_lt_one (Int.coe_nat_pos.2 q.pos), mul_one,
zero_mul] at hq | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· rw [abs_of_nonpos hq]
rw [← @num_den q, ← divInt_zero_one, Rat.le_def (Int.coe_nat_pos.2 q.pos) zero_lt_one, mul_one,
zero_mul] at hq
rw [Int.ofNat_natAbs_of_nonpos hq, ← neg_def, num_den]
· rw [abs_of_non... | Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
case inr
a b c q : ℚ
hq : 0 ≤ q.num
⊢ q = ↑(Int.natAbs q.num) /. ↑q.den | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Rat.Basic
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.rat.order from "leanprov... | rw [Int.natAbs_of_nonneg hq, num_den] | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den := by
rcases le_total q 0 with hq | hq
· rw [abs_of_nonpos hq]
rw [← @num_den q, ← divInt_zero_one, Rat.le_def (Int.coe_nat_pos.2 q.pos) zero_lt_one, mul_one,
zero_mul] at hq
rw [Int.ofNat_natAbs_of_nonpos hq, ← neg_def, num_den]
· rw [abs_of_non... | Mathlib.Data.Rat.Order.298_0.NTjR6KCugNscheB | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | Mathlib_Data_Rat_Order |
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
⊢ ∃ Y, Simple (underlying.obj Y) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by
| Mathlib.CategoryTheory.Noetherian.82_0.bQzOkYhrFw250Z9 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) | Mathlib_CategoryTheory_Noetherian |
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
this : Nontrivial (Subobject X)
⊢ ∃ Y, Simple (underlying.obj Y) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X) | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
| Mathlib.CategoryTheory.Noetherian.82_0.bQzOkYhrFw250Z9 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) | Mathlib_CategoryTheory_Noetherian |
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
this✝ : Nontrivial (Subobject X)
this : IsAtomic (Subobject X)
⊢ ∃ Y, Simple (underlying.obj Y) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
| Mathlib.CategoryTheory.Noetherian.82_0.bQzOkYhrFw250Z9 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) | Mathlib_CategoryTheory_Noetherian |
case intro
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
this✝ : Nontrivial (Subobject X)
this : IsAtomic (Subobject X)
Y : Subobject X
s : IsAtom Y ∧ Y ≤ ⊤
⊢ ∃ Y, Simple (underlying.obj Y) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩ | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exis... | Mathlib.CategoryTheory.Noetherian.82_0.bQzOkYhrFw250Z9 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) | Mathlib_CategoryTheory_Noetherian |
C : Type u_1
inst✝³ : Category.{?u.7072, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
⊢ Mono (simpleSubobjectArrow h) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | dsimp only [simpleSubobjectArrow] | instance mono_simpleSubobjectArrow {X : C} [ArtinianObject X] (h : ¬IsZero X) :
Mono (simpleSubobjectArrow h) := by
| Mathlib.CategoryTheory.Noetherian.101_0.bQzOkYhrFw250Z9 | instance mono_simpleSubobjectArrow {X : C} [ArtinianObject X] (h : ¬IsZero X) :
Mono (simpleSubobjectArrow h) | Mathlib_CategoryTheory_Noetherian |
C : Type u_1
inst✝³ : Category.{?u.7072, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
⊢ Mono (arrow (Exists.choose (_ : ∃ Y, Simple (underlying.obj Y)))) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noe... | infer_instance | instance mono_simpleSubobjectArrow {X : C} [ArtinianObject X] (h : ¬IsZero X) :
Mono (simpleSubobjectArrow h) := by
dsimp only [simpleSubobjectArrow]
| Mathlib.CategoryTheory.Noetherian.101_0.bQzOkYhrFw250Z9 | instance mono_simpleSubobjectArrow {X : C} [ArtinianObject X] (h : ¬IsZero X) :
Mono (simpleSubobjectArrow h) | Mathlib_CategoryTheory_Noetherian |
n : ℕ
⊢ ack 0 n = n + 1 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack] | @[simp]
theorem ack_zero (n : ℕ) : ack 0 n = n + 1 := by | Mathlib.Computability.Ackermann.69_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_zero (n : ℕ) : ack 0 n = n + 1 | Mathlib_Computability_Ackermann |
m : ℕ
⊢ ack (m + 1) 0 = ack m 1 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack] | @[simp]
theorem ack_succ_zero (m : ℕ) : ack (m + 1) 0 = ack m 1 := by | Mathlib.Computability.Ackermann.73_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_succ_zero (m : ℕ) : ack (m + 1) 0 = ack m 1 | Mathlib_Computability_Ackermann |
m n : ℕ
⊢ ack (m + 1) (n + 1) = ack m (ack (m + 1) n) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack] | @[simp]
theorem ack_succ_succ (m n : ℕ) : ack (m + 1) (n + 1) = ack m (ack (m + 1) n) := by | Mathlib.Computability.Ackermann.77_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_succ_succ (m n : ℕ) : ack (m + 1) (n + 1) = ack m (ack (m + 1) n) | Mathlib_Computability_Ackermann |
n : ℕ
⊢ ack 1 n = n + 2 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | induction' n with n IH | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 := by
| Mathlib.Computability.Ackermann.81_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 | Mathlib_Computability_Ackermann |
case zero
⊢ ack 1 zero = zero + 2 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rfl | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 := by
induction' n with n IH
· | Mathlib.Computability.Ackermann.81_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 1 n = n + 2
⊢ ack 1 (succ n) = succ n + 2 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | simp [IH] | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 := by
induction' n with n IH
· rfl
· | Mathlib.Computability.Ackermann.81_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_one (n : ℕ) : ack 1 n = n + 2 | Mathlib_Computability_Ackermann |
n : ℕ
⊢ ack 2 n = 2 * n + 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | induction' n with n IH | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by
| Mathlib.Computability.Ackermann.88_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 | Mathlib_Computability_Ackermann |
case zero
⊢ ack 2 zero = 2 * zero + 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rfl | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by
induction' n with n IH
· | Mathlib.Computability.Ackermann.88_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 2 n = 2 * n + 3
⊢ ack 2 (succ n) = 2 * succ n + 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | simpa [mul_succ] | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by
induction' n with n IH
· rfl
· | Mathlib.Computability.Ackermann.88_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 | Mathlib_Computability_Ackermann |
n : ℕ
⊢ ack 3 n = 2 ^ (n + 3) - 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | induction' n with n IH | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
| Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
case zero
⊢ ack 3 zero = 2 ^ (zero + 3) - 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rfl | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
⊢ ack 3 (succ n) = 2 ^ (succ n + 3) - 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right] | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_num | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
| Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
⊢ 2 * 3 ≤ 2 * 2 ^ 3 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | norm_num | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
have H : 2 * 3 ≤ 2 * 2 ^ 3 := by | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
H : 2 * 3 ≤ 2 * 2 ^ 3
⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | apply H.trans | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_n... | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
case succ
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
H : 2 * 3 ≤ 2 * 2 ^ 3
⊢ 2 * 2 ^ 3 ≤ 2 * 2 ^ (n + 3) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | simp [pow_le_pow_right (show 1 ≤ 2 by norm_num)] | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_n... | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
n : ℕ
IH : ack 3 n = 2 ^ (n + 3) - 3
H : 2 * 3 ≤ 2 * 2 ^ 3
⊢ 1 ≤ 2 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | norm_num | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_n... | Mathlib.Computability.Ackermann.96_0.nk1BuTevlQj1gCN | @[simp]
theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 | Mathlib_Computability_Ackermann |
n : ℕ
⊢ 0 < ack 0 n | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | simp | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by | Mathlib.Computability.Ackermann.107_0.nk1BuTevlQj1gCN | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos | Mathlib_Computability_Ackermann |
m : ℕ
⊢ 0 < ack (m + 1) 0 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack_succ_zero] | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
| Mathlib.Computability.Ackermann.107_0.nk1BuTevlQj1gCN | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos | Mathlib_Computability_Ackermann |
m : ℕ
⊢ 0 < ack m 1 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | apply ack_pos | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
| Mathlib.Computability.Ackermann.107_0.nk1BuTevlQj1gCN | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos | Mathlib_Computability_Ackermann |
m n : ℕ
⊢ 0 < ack (m + 1) (n + 1) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack_succ_succ] | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
| Mathlib.Computability.Ackermann.107_0.nk1BuTevlQj1gCN | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos | Mathlib_Computability_Ackermann |
m n : ℕ
⊢ 0 < ack m (ack (m + 1) n) | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | apply ack_pos | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
| Mathlib.Computability.Ackermann.107_0.nk1BuTevlQj1gCN | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos | Mathlib_Computability_Ackermann |
n : ℕ
⊢ 1 < ack (0 + 1) n | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | simp | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by | Mathlib.Computability.Ackermann.117_0.nk1BuTevlQj1gCN | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply one_lt_ack_succ_left
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply one_lt_ack_succ_left | Mathlib_Computability_Ackermann |
m : ℕ
⊢ 1 < ack (m + 1 + 1) 0 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | rw [ack_succ_zero] | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by simp
| m + 1, 0 => by
| Mathlib.Computability.Ackermann.117_0.nk1BuTevlQj1gCN | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply one_lt_ack_succ_left
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply one_lt_ack_succ_left | Mathlib_Computability_Ackermann |
m : ℕ
⊢ 1 < ack (m + 1) 1 | /-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "le... | apply one_lt_ack_succ_left | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
| Mathlib.Computability.Ackermann.117_0.nk1BuTevlQj1gCN | theorem one_lt_ack_succ_left : ∀ m n, 1 < ack (m + 1) n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply one_lt_ack_succ_left
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply one_lt_ack_succ_left | Mathlib_Computability_Ackermann |
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