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 |
|---|---|---|---|---|---|---|
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : z ⊓ (x ⊓ y) = x \ y ⊓ (x ⊓ y)
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | exact (eq_of_inf_eq_sup_eq i s).symm | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]
rw [inf_comm] at i
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y \ x ⊔ x = y \ x ⊔ (x ⊔ x ⊓ y) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_self] | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x :=
calc
y \ x ⊔ x = y \ x ⊔ (x ⊔ x ⊓ y) := by | Mathlib.Order.BooleanAlgebra.142_0.ewE75DLNneOU8G5 | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y \ x ⊔ (x ⊔ x ⊓ y) = y ⊓ x ⊔ y \ x ⊔ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x :=
calc
y \ x ⊔ x = y \ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]
_ = y ⊓ x ⊔ y \ x ⊔ x := by | Mathlib.Order.BooleanAlgebra.142_0.ewE75DLNneOU8G5 | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊓ x ⊔ y \ x ⊔ x = y ⊔ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff] | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x :=
calc
y \ x ⊔ x = y \ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]
_ = y ⊓ x ⊔ y \ x ⊔ x := by ac_rfl
_ = y ⊔ x := by | Mathlib.Order.BooleanAlgebra.142_0.ewE75DLNneOU8G5 | private theorem sdiff_sup_self' : y \ x ⊔ x = y ⊔ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ ⊥ = x ⊓ y ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_inf_sdiff] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y ⊓ x \ y = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_left] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y = (y ⊓ x ⊔ x ⊓ y \ x) ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_idem] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ x ⊔ x ⊓ y \ x) ⊓ x \ y = x ⊓ y ⊓ x \ y ⊔ x ⊓ y \ x ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_right, @inf_comm _ _ x y] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y ⊓ x \ y ⊔ x ⊓ y \ x ⊓ x \ y = x ⊓ y \ x ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_inf_sdiff, bot_sup_eq] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ x ⊓ x \ y = x ⊓ x \ y ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ x \ y ⊓ y \ x = x \ y ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_of_le_right sdiff_le'] | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.148_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ x = (x ⊓ y ⊔ x \ y) ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff] | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ :=
calc
x ⊓ y \ x = (x ⊓ y ⊔ x \ y) ⊓ y \ x := by | Mathlib.Order.BooleanAlgebra.167_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊓ y ⊔ x \ y) ⊓ y \ x = x ⊓ y ⊓ y \ x ⊔ x \ y ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_right] | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ :=
calc
x ⊓ y \ x = (x ⊓ y ⊔ x \ y) ⊓ y \ x := by rw [sup_inf_sdiff]
_ = x ⊓ y ⊓ y \ x ⊔ x \ y ⊓ y \ x := by | Mathlib.Order.BooleanAlgebra.167_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y ⊓ y \ x ⊔ x \ y ⊓ y \ x = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [@inf_comm _ _ x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq] | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ :=
calc
x ⊓ y \ x = (x ⊓ y ⊔ x \ y) ⊓ y \ x := by rw [sup_inf_sdiff]
_ = x ⊓ y ⊓ y \ x ⊔ x \ y ⊓ y \ x := by rw [inf_sup_right]
_ = ⊥ := by | Mathlib.Order.BooleanAlgebra.167_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_self_right : x ⊓ y \ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y \ x ⊓ x = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_comm, inf_sdiff_self_right] | @[simp]
theorem inf_sdiff_self_left : y \ x ⊓ x = ⊥ := by | Mathlib.Order.BooleanAlgebra.175_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_self_left : y \ x ⊓ x = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z✝ : α
inst✝ : GeneralizedBooleanAlgebra α
src✝¹ : GeneralizedBooleanAlgebra α := inst✝
src✝ : OrderBot α := toOrderBot
y x z : α
h : y \ x ≤ z
⊢ y \ x = x ⊓ y \ x ⊔ z ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_eq_right.2 h, inf_sdiff_self_right, bot_sup_eq] | instance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :
GeneralizedCoheytingAlgebra α :=
{ ‹GeneralizedBooleanAlgebra α›, GeneralizedBooleanAlgebra.toOrderBot with
sdiff := (· \ ·),
sdiff_le_iff := fun y x z =>
⟨fun h =>
le_of_inf_le_sup_le
(le_of_eq
... | Mathlib.Order.BooleanAlgebra.180_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z✝ : α
inst✝ : GeneralizedBooleanAlgebra α
src✝¹ : GeneralizedBooleanAlgebra α := inst✝
src✝ : OrderBot α := toOrderBot
y x z : α
h : y \ x ≤ z
⊢ y ⊔ (x ⊔ z) = y \ x ⊔ x ⊔ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_assoc, ← @sdiff_sup_self' _ x y] | instance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :
GeneralizedCoheytingAlgebra α :=
{ ‹GeneralizedBooleanAlgebra α›, GeneralizedBooleanAlgebra.toOrderBot with
sdiff := (· \ ·),
sdiff_le_iff := fun y x z =>
⟨fun h =>
le_of_inf_le_sup_le
(le_of_eq
... | Mathlib.Order.BooleanAlgebra.180_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z✝ : α
inst✝ : GeneralizedBooleanAlgebra α
src✝¹ : GeneralizedBooleanAlgebra α := inst✝
src✝ : OrderBot α := toOrderBot
y x z : α
h : y \ x ≤ z
⊢ y \ x ⊔ x ⊔ z = x ⊔ z ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | instance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :
GeneralizedCoheytingAlgebra α :=
{ ‹GeneralizedBooleanAlgebra α›, GeneralizedBooleanAlgebra.toOrderBot with
sdiff := (· \ ·),
sdiff_le_iff := fun y x z =>
⟨fun h =>
le_of_inf_le_sup_le
(le_of_eq
... | Mathlib.Order.BooleanAlgebra.180_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z✝ : α
inst✝ : GeneralizedBooleanAlgebra α
src✝¹ : GeneralizedBooleanAlgebra α := inst✝
src✝ : OrderBot α := toOrderBot
y x z : α
h : y ≤ x ⊔ z
⊢ x ⊔ z ⊔ x ≤ z ⊔ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_assoc, sup_comm, sup_assoc, sup_idem] | instance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :
GeneralizedCoheytingAlgebra α :=
{ ‹GeneralizedBooleanAlgebra α›, GeneralizedBooleanAlgebra.toOrderBot with
sdiff := (· \ ·),
sdiff_le_iff := fun y x z =>
⟨fun h =>
le_of_inf_le_sup_le
(le_of_eq
... | Mathlib.Order.BooleanAlgebra.180_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x ≤ y ∧ Disjoint x z
⊢ x ≤ y \ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← h.2.sdiff_eq_left] | lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z :=
⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦
by | Mathlib.Order.BooleanAlgebra.217_0.ewE75DLNneOU8G5 | lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x ≤ y ∧ Disjoint x z
⊢ x \ z ≤ y \ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | exact sdiff_le_sdiff_right h.1 | lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z :=
⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦
by rw [← h.2.sdiff_eq_left]; | Mathlib.Order.BooleanAlgebra.217_0.ewE75DLNneOU8G5 | lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hi : Disjoint x z
hs : x ⊔ z = y
h : y ⊓ x = x
⊢ y ⊓ x ⊔ z = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [h, hs] | theorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z :=
have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le
sdiff_unique (by | Mathlib.Order.BooleanAlgebra.228_0.ewE75DLNneOU8G5 | theorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hi : Disjoint x z
hs : x ⊔ z = y
h : y ⊓ x = x
⊢ y ⊓ x ⊓ z = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [h, hi.eq_bot] | theorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z :=
have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le
sdiff_unique (by rw [h, hs]) (by | Mathlib.Order.BooleanAlgebra.228_0.ewE75DLNneOU8G5 | theorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hd : Disjoint x z
hz : z ≤ y
hs : y ≤ x ⊔ z
⊢ y ⊓ x ⊔ z = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_eq_right] at hs | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z :=
sdiff_unique
(by
| Mathlib.Order.BooleanAlgebra.233_0.ewE75DLNneOU8G5 | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hd : Disjoint x z
hz : z ≤ y
hs : (x ⊔ z) ⊓ y = y
⊢ y ⊓ x ⊔ z = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rwa [sup_inf_right, inf_sup_right, @sup_comm _ _ x, inf_sup_self, inf_comm, @sup_comm _ _ z,
hs, sup_eq_left] | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z :=
sdiff_unique
(by
rw [← inf_eq_right] at hs
| Mathlib.Order.BooleanAlgebra.233_0.ewE75DLNneOU8G5 | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hd : Disjoint x z
hz : z ≤ y
hs : y ≤ x ⊔ z
⊢ y ⊓ x ⊓ z = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_assoc, hd.eq_bot, inf_bot_eq] | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z :=
sdiff_unique
(by
rw [← inf_eq_right] at hs
rwa [sup_inf_right, inf_sup_right, @sup_comm _ _ x, inf_sup_self, inf_comm, @sup_comm _ _ z,
hs, sup_eq_left])
(by | Mathlib.Order.BooleanAlgebra.233_0.ewE75DLNneOU8G5 | protected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :
y \ x = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : Disjoint z (y \ x)
⊢ z ⊔ y \ x ≤ x ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_sdiff_cancel_right hx] | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x :=
⟨fun H =>
le_of_inf_le_sup_le (le_trans H.le_bot bot_le)
(by
| Mathlib.Order.BooleanAlgebra.244_0.ewE75DLNneOU8G5 | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : Disjoint z (y \ x)
⊢ z ⊔ y \ x ≤ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | refine' le_trans (sup_le_sup_left sdiff_le z) _ | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x :=
⟨fun H =>
le_of_inf_le_sup_le (le_trans H.le_bot bot_le)
(by
rw [sup_sdiff_cancel_right hx]
| Mathlib.Order.BooleanAlgebra.244_0.ewE75DLNneOU8G5 | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : Disjoint z (y \ x)
⊢ z ⊔ y ≤ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_eq_right.2 hz] | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x :=
⟨fun H =>
le_of_inf_le_sup_le (le_trans H.le_bot bot_le)
(by
rw [sup_sdiff_cancel_right hx]
refine' le_trans (sup_le_sup_left sdiff_le z) _
| Mathlib.Order.BooleanAlgebra.244_0.ewE75DLNneOU8G5 | theorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
⊢ z ⊓ y \ x = ⊥ ↔ z ≤ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← disjoint_iff] | theorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x := by
| Mathlib.Order.BooleanAlgebra.260_0.ewE75DLNneOU8G5 | theorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
⊢ Disjoint z (y \ x) ↔ z ≤ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | exact disjoint_sdiff_iff_le hz hx | theorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x := by
rw [← disjoint_iff]
| Mathlib.Order.BooleanAlgebra.260_0.ewE75DLNneOU8G5 | theorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ y = z ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | apply le_antisymm | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
| Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ y ≤ z ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | conv_lhs => rw [← sup_inf_sdiff y x] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff y x] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff y x] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff y x] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ y ⊓ x ⊔ y \ x ≤ z ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | apply sup_le_sup_right | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
| Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a.h₁
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ y ⊓ x ≤ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rwa [inf_eq_right.2 hx] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
| Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ z ⊔ y \ x ≤ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | apply le_trans | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a.a
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ z ⊔ y \ x ≤ ?a.b✝ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | apply sup_le_sup_right hz | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
case a.a
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ≤ z
⊢ y ⊔ y \ x ≤ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_sdiff_left] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : y = z ⊔ y \ x
⊢ x ≤ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | conv_lhs at H => rw [← sup_sdiff_cancel_right hx] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : y = z ⊔ y \ x
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_sdiff_cancel_right hx] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : y = z ⊔ y \ x
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_sdiff_cancel_right hx] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : y = z ⊔ y \ x
| y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_sdiff_cancel_right hx] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ⊔ y \ x = z ⊔ y \ x
⊢ x ≤ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | refine' le_of_inf_le_sup_le _ H.le | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ⊔ y \ x = z ⊔ y \ x
⊢ x ⊓ y \ x ≤ z ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sdiff_self_right] | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hz : z ≤ y
hx : x ≤ y
H : x ⊔ y \ x = z ⊔ y \ x
⊢ ⊥ ≤ z ⊓ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | exact bot_le | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x :=
⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
fun H => by
... | Mathlib.Order.BooleanAlgebra.266_0.ewE75DLNneOU8G5 | theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_left] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [@inf_sup_left _ _ y] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \ z)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \ z)) = (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff, sup_inf_sdiff] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) = (y ⊔ y ⊓ z) ⊓ (y ⊔ y ⊓ x) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊔ y ⊓ z) ⊓ (y ⊔ y ⊓ x) = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_self, sup_inf_self, inf_idem] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊓ (x ⊔ z) ⊓ (y \ x ⊓ y \ z) = (y ⊓ x ⊔ y ⊓ z) ⊓ (y \ x ⊓ y \ z) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_left] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ x ⊔ y ⊓ z) ⊓ (y \ x ⊓ y \ z) = y ⊓ x ⊓ (y \ x ⊓ y \ z) ⊔ y ⊓ z ⊓ (y \ x ⊓ y \ z) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_right] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊓ x ⊓ (y \ x ⊓ y \ z) ⊔ y ⊓ z ⊓ (y \ x ⊓ y \ z) = y ⊓ x ⊓ y \ x ⊓ y \ z ⊔ y \ x ⊓ (y \ z ⊓ (y ⊓ z)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊓ x ⊓ y \ x ⊓ y \ z ⊔ y \ x ⊓ (y \ z ⊓ (y ⊓ z)) = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_inf_sdiff, bot_inf_eq, bot_sup_eq, @inf_comm _ _ (y \ z),
inf_inf_sdiff, inf_bot_eq] | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) :=
by rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓... | Mathlib.Order.BooleanAlgebra.283_0.ewE75DLNneOU8G5 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y \ x = y \ z
⊢ y ⊓ x ⊓ ?m.20748 h = y ⊓ z ⊓ ?m.20748 h | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_inf_sdiff, h, inf_inf_sdiff] | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z :=
⟨fun h => eq_of_inf_eq_sup_eq (by | Mathlib.Order.BooleanAlgebra.301_0.ewE75DLNneOU8G5 | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y \ x = y \ z
⊢ y ⊓ x ⊔ y \ x = y ⊓ z ⊔ y \ x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff, h, sup_inf_sdiff] | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z :=
⟨fun h => eq_of_inf_eq_sup_eq (by rw [inf_inf_sdiff, h, inf_inf_sdiff])
(by | Mathlib.Order.BooleanAlgebra.301_0.ewE75DLNneOU8G5 | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y ⊓ x = y ⊓ z
⊢ y \ x = y \ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sdiff_inf_self_right, ← sdiff_inf_self_right z y, inf_comm, h, inf_comm] | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z :=
⟨fun h => eq_of_inf_eq_sup_eq (by rw [inf_inf_sdiff, h, inf_inf_sdiff])
(by rw [sup_inf_sdiff, h, sup_inf_sdiff]),
fun h => by | Mathlib.Order.BooleanAlgebra.301_0.ewE75DLNneOU8G5 | theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ y = x ↔ x \ y = x \ ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_bot] | theorem sdiff_eq_self_iff_disjoint : x \ y = x ↔ Disjoint y x :=
calc
x \ y = x ↔ x \ y = x \ ⊥ := by | Mathlib.Order.BooleanAlgebra.307_0.ewE75DLNneOU8G5 | theorem sdiff_eq_self_iff_disjoint : x \ y = x ↔ Disjoint y x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y = x ⊓ ⊥ ↔ Disjoint y x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_bot_eq, inf_comm, disjoint_iff] | theorem sdiff_eq_self_iff_disjoint : x \ y = x ↔ Disjoint y x :=
calc
x \ y = x ↔ x \ y = x \ ⊥ := by rw [sdiff_bot]
_ ↔ x ⊓ y = x ⊓ ⊥ := sdiff_eq_sdiff_iff_inf_eq_inf
_ ↔ Disjoint y x := by | Mathlib.Order.BooleanAlgebra.307_0.ewE75DLNneOU8G5 | theorem sdiff_eq_self_iff_disjoint : x \ y = x ↔ Disjoint y x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ y = x ↔ Disjoint x y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_eq_self_iff_disjoint, disjoint_comm] | theorem sdiff_eq_self_iff_disjoint' : x \ y = x ↔ Disjoint x y := by
| Mathlib.Order.BooleanAlgebra.314_0.ewE75DLNneOU8G5 | theorem sdiff_eq_self_iff_disjoint' : x \ y = x ↔ Disjoint x y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hx : y ≤ x
hy : y ≠ ⊥
⊢ x \ y < x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | refine' sdiff_le.lt_of_ne fun h => hy _ | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x := by
| Mathlib.Order.BooleanAlgebra.318_0.ewE75DLNneOU8G5 | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hx : y ≤ x
hy : y ≠ ⊥
h : x \ y = x
⊢ y = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x := by
refine' sdiff_le.lt_of_ne fun h => hy _
| Mathlib.Order.BooleanAlgebra.318_0.ewE75DLNneOU8G5 | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hx : y ≤ x
hy : y ≠ ⊥
h : x ⊓ y = ⊥
⊢ y = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← h, inf_eq_right.mpr hx] | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x := by
refine' sdiff_le.lt_of_ne fun h => hy _
rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h
| Mathlib.Order.BooleanAlgebra.318_0.ewE75DLNneOU8G5 | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y ⊓ z ⊔ y \ z = x ⊓ (y ⊓ z) ⊔ y \ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_assoc] | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z :=
calc
x ⊓ y ⊓ z ⊔ y \ z = x ⊓ (y ⊓ z) ⊔ y \ z := by | Mathlib.Order.BooleanAlgebra.334_0.ewE75DLNneOU8G5 | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y ⊓ z) ⊔ y \ z = (x ⊔ y \ z) ⊓ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_right, sup_inf_sdiff] | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z :=
calc
x ⊓ y ⊓ z ⊔ y \ z = x ⊓ (y ⊓ z) ⊔ y \ z := by rw [inf_assoc]
_ = (x ⊔ y \ z) ⊓ y := by | Mathlib.Order.BooleanAlgebra.334_0.ewE75DLNneOU8G5 | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊔ y \ z) ⊓ y = x ⊓ y ⊔ y \ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_right, inf_sdiff_left] | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z :=
calc
x ⊓ y ⊓ z ⊔ y \ z = x ⊓ (y ⊓ z) ⊔ y \ z := by rw [inf_assoc]
_ = (x ⊔ y \ z) ⊓ y := by rw [sup_inf_right, sup_inf_sdiff]
_ = x ⊓ y ⊔ y \ z := by | Mathlib.Order.BooleanAlgebra.334_0.ewE75DLNneOU8G5 | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
| Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ (y \ z) = z ⊓ x ⊔ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | apply sdiff_unique | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
| Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
case s
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓ z ⊔ x \ y)) := by ac_rfl
_ = x ⊓ (y \ z ⊔ x ⊓ z ⊔ x \ y) := by rw [sup_inf_self, sup_sdiff_left, ← sup_assoc]
_ = x ⊓ (y \ z ⊓ (z ⊔ y) ⊔... | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_right] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓ z ⊔ x \ y)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓ z ⊔ x \ y)) = x ⊓ (y \ z ⊔ x ⊓ z ⊔ x \ y) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_self, sup_sdiff_left, ← sup_assoc] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊔ x ⊓ z ⊔ x \ y) = x ⊓ (y \ z ⊓ (z ⊔ y) ⊔ x ⊓ (z ⊔ y) ⊔ x \ y) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_left, sdiff_sup_self', inf_sup_right, @sup_comm _ _ y] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊓ (z ⊔ y) ⊔ x ⊓ (z ⊔ y) ⊔ x \ y) = x ⊓ (y \ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \ y) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sdiff_sup_right, @inf_sup_left _ _ x z y] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \ y) = x ⊓ (y \ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \ y))) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \ y))) = x ⊓ (y \ z ⊔ (x ⊔ x ⊓ z)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_sdiff, @sup_comm _ _ (x ⊓ z)] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊔ (x ⊔ x ⊓ z)) = x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_self, sup_comm, inf_sup_self] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
case i
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | calc
x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = x ⊓ y \ z ⊓ (z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y := by rw [inf_sup_left]
_ = x ⊓ (y \ z ⊓ z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y := by ac_rfl
_ = x ⊓ y \ z ⊓ x \ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]
_ = x ⊓ (y \ z ⊓ y) ⊓ x \ y := by conv_lhs => rw [←... | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = x ⊓ y \ z ⊓ (z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sup_left] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ (z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y = x ⊓ (y \ z ⊓ z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊓ z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y = x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ x \ y = x ⊓ (y \ z ⊓ y) ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | conv_lhs => rw [← inf_sdiff_left] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
| x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_sdiff_left] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
| x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_sdiff_left] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
| x ⊓ y \ z ⊓ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_sdiff_left] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊓ y) ⊓ x \ y = x ⊓ (y \ z ⊓ (y ⊓ x \ y)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ (y \ z ⊓ (y ⊓ x \ y)) = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq] | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) :=
by rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓... | Mathlib.Order.BooleanAlgebra.341_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ y ⊔ x ⊓ y ⊓ z = z ⊓ x ⊓ y ⊔ x \ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | ac_rfl | theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z :=
calc
x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := sdiff_sdiff_right
_ = z ⊓ x ⊓ y ⊔ x \ y := by | Mathlib.Order.BooleanAlgebra.365_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z ⊓ x ⊓ y ⊔ x \ y = x \ y ⊔ x ⊓ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_inf_inf_sdiff, sup_comm, inf_comm] | theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z :=
calc
x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := sdiff_sdiff_right
_ = z ⊓ x ⊓ y ⊔ x \ y := by ac_rfl
_ = x \ y ⊔ x ⊓ z := by | Mathlib.Order.BooleanAlgebra.365_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : z ≤ x
⊢ x \ (y \ z) = x \ y ⊔ z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_sdiff_right', inf_eq_right.2 h] | theorem sdiff_sdiff_eq_sdiff_sup (h : z ≤ x) : x \ (y \ z) = x \ y ⊔ z := by
| Mathlib.Order.BooleanAlgebra.372_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_eq_sdiff_sup (h : z ≤ x) : x \ (y \ z) = x \ y ⊔ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ (x \ y) = x ⊓ y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_sdiff_right, inf_idem, sdiff_self, bot_sup_eq] | @[simp]
theorem sdiff_sdiff_right_self : x \ (x \ y) = x ⊓ y := by
| Mathlib.Order.BooleanAlgebra.376_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_sdiff_right_self : x \ (x \ y) = x ⊓ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y ≤ x
⊢ x \ (x \ y) = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sdiff_sdiff_right_self, inf_of_le_right h] | theorem sdiff_sdiff_eq_self (h : y ≤ x) : x \ (x \ y) = y := by
| Mathlib.Order.BooleanAlgebra.381_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_eq_self (h : y ≤ x) : x \ (x \ y) = y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hy : y ≤ x
h : x \ y = z
⊢ x \ z = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← h, sdiff_sdiff_eq_self hy] | theorem sdiff_eq_symm (hy : y ≤ x) (h : x \ y = z) : x \ z = y := by
| Mathlib.Order.BooleanAlgebra.385_0.ewE75DLNneOU8G5 | theorem sdiff_eq_symm (hy : y ≤ x) (h : x \ y = z) : x \ z = y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hxz : x ≤ z
hyz : y ≤ z
h : z \ x = z \ y
⊢ x = y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sdiff_sdiff_eq_self hxz, h, sdiff_sdiff_eq_self hyz] | theorem eq_of_sdiff_eq_sdiff (hxz : x ≤ z) (hyz : y ≤ z) (h : z \ x = z \ y) : x = y := by
| Mathlib.Order.BooleanAlgebra.393_0.ewE75DLNneOU8G5 | theorem eq_of_sdiff_eq_sdiff (hxz : x ≤ z) (hyz : y ≤ z) (h : z \ x = z \ y) : x = y | Mathlib_Order_BooleanAlgebra |
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