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
⊢ (x \ y) \ z = 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 [sdiff_sdiff_left, sdiff_sup] | theorem sdiff_sdiff_left' : (x \ y) \ z = x \ y ⊓ x \ z := by | Mathlib.Order.BooleanAlgebra.397_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_left' : (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 ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (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... | rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right] | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) :=
by | Mathlib.Order.BooleanAlgebra.400_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) = z ⊓ (z \ x ⊔ y) ⊓ (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... | rw [sup_inf_left, sup_comm, sup_inf_sdiff] | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) :=
by rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]
_ = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by | Mathlib.Order.BooleanAlgebra.400_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) = z ⊓ (z \ x ⊔ y) ⊓ (z ⊓ (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_left, @sup_comm _ _ (z \ y), sup_inf_sdiff] | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) :=
by rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]
_ = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by rw [sup_inf_left, sup_comm, sup_inf_sdi... | Mathlib.Order.BooleanAlgebra.400_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z ⊓ (z \ x ⊔ y) ⊓ (z ⊓ (z \ y ⊔ x)) = z ⊓ z ⊓ (z \ 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... | ac_rfl | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) :=
by rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]
_ = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by rw [sup_inf_left, sup_comm, sup_inf_sdi... | Mathlib.Order.BooleanAlgebra.400_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z ⊓ z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) = z ⊓ (z \ 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 [inf_idem] | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) :=
by rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]
_ = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by rw [sup_inf_left, sup_comm, sup_inf_sdi... | Mathlib.Order.BooleanAlgebra.400_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z \ (x \ y) ⊓ z \ (y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (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... | rw [sdiff_sdiff_right, sdiff_sdiff_right] | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y :=
calc
z \ (x \ y ⊔ y \ x) = z \ (x \ y) ⊓ z \ (y \ x) := sdiff_sup
_ = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by | Mathlib.Order.BooleanAlgebra.411_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) = (z \ x ⊔ z ⊓ y ⊓ 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... | ac_rfl | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y :=
calc
z \ (x \ y ⊔ y \ x) = z \ (x \ y) ⊓ z \ (y \ x) := sdiff_sup
_ = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by rw [sdiff_sdiff_right, sdiff_sdiff_right]
_ = (z \ x ⊔ z ⊓ y ⊓ x) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by | Mathlib.Order.BooleanAlgebra.411_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ z \ x ⊓ z \ y ⊔ z ⊓ y ⊓ x = z ⊓ 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... | ac_rfl | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y :=
calc
z \ (x \ y ⊔ y \ x) = z \ (x \ y) ⊓ z \ (y \ x) := sdiff_sup
_ = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by rw [sdiff_sdiff_right, sdiff_sdiff_right]
_ = (z \ x ⊔ z ⊓ y ⊓ x) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by ac_rfl
... | Mathlib.Order.BooleanAlgebra.411_0.ewE75DLNneOU8G5 | theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hcb : z ≤ y
⊢ (x \ z) \ (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 [le_antisymm_iff, sdiff_le_comm] | lemma sdiff_sdiff_sdiff_cancel_right (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y := by
| Mathlib.Order.BooleanAlgebra.424_0.ewE75DLNneOU8G5 | lemma sdiff_sdiff_sdiff_cancel_right (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
hcb : z ≤ y
⊢ (x \ z) \ (x \ y) ≤ y \ z ∧ 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... | exact ⟨sdiff_sdiff_sdiff_le_sdiff,
(disjoint_sdiff_self_left.mono_right sdiff_le).le_sdiff_of_le_left <| sdiff_le_sdiff_left hcb⟩ | lemma sdiff_sdiff_sdiff_cancel_right (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y := by
rw [le_antisymm_iff, sdiff_le_comm]
| Mathlib.Order.BooleanAlgebra.424_0.ewE75DLNneOU8G5 | lemma sdiff_sdiff_sdiff_cancel_right (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ 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 [sup_inf_left] | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ 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 ⊓ z ⊔ y \ z) = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ 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 [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right] | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ 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 ⊓ y ⊓ z ⊔ y \ z) = (y ⊓ (x ⊓ (x ⊔ z)) ⊔ x \ 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... | ac_rfl | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_... | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ (x ⊓ (x ⊔ z)) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) = (y ⊓ x ⊔ x \ z) ⊓ (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_self, sup_inf_inf_sdiff] | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_... | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ x ⊔ x \ z) ⊓ (x ⊓ y ⊔ y \ z) = 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 [@inf_comm _ _ y, sup_inf_left] | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_... | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ 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 ⊓ y \ z) = x ⊓ y ⊓ (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... | ac_rfl | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_... | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ 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) ⊓ 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_sdiff_self_right, inf_bot_eq, bot_inf_eq] | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x \ z ⊓ y \ z = (x ⊓ y ⊓ z ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) := by rw [sup_inf_left]
_ = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \ z) ⊓ (x ⊓ y ⊓ z ⊔ y \ z) :=
by rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_... | Mathlib.Order.BooleanAlgebra.429_0.ewE75DLNneOU8G5 | theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ 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 \ z = 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 [inf_assoc] | theorem inf_sdiff_assoc : (x ⊓ y) \ z = x ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x ⊓ y \ z = x ⊓ (y ⊓ z) ⊔ x ⊓ y \ z := by | Mathlib.Order.BooleanAlgebra.444_0.ewE75DLNneOU8G5 | theorem inf_sdiff_assoc : (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 ⊓ (y ⊓ z ⊔ 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 [sup_inf_sdiff] | theorem inf_sdiff_assoc : (x ⊓ y) \ z = x ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x ⊓ y \ z = x ⊓ (y ⊓ z) ⊔ x ⊓ y \ z := by rw [inf_assoc]
_ = x ⊓ (y ⊓ z ⊔ y \ z) := inf_sup_left.symm
_ = x ⊓ y := by | Mathlib.Order.BooleanAlgebra.444_0.ewE75DLNneOU8G5 | theorem inf_sdiff_assoc : (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 ⊓ y ⊓ z ⊓ (x ⊓ y \ z) = x ⊓ 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 inf_sdiff_assoc : (x ⊓ y) \ z = x ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x ⊓ y \ z = x ⊓ (y ⊓ z) ⊔ x ⊓ y \ z := by rw [inf_assoc]
_ = x ⊓ (y ⊓ z ⊔ y \ z) := inf_sup_left.symm
_ = x ⊓ y := by rw [sup_inf_sdiff])
(calc
x ⊓ y ⊓ z ⊓ (x ⊓ y \ z) = x ⊓ x ⊓ (y ⊓ z ⊓ y \ z) := by | Mathlib.Order.BooleanAlgebra.444_0.ewE75DLNneOU8G5 | theorem inf_sdiff_assoc : (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 ⊓ 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, inf_bot_eq] | theorem inf_sdiff_assoc : (x ⊓ y) \ z = x ⊓ y \ z :=
sdiff_unique
(calc
x ⊓ y ⊓ z ⊔ x ⊓ y \ z = x ⊓ (y ⊓ z) ⊔ x ⊓ y \ z := by rw [inf_assoc]
_ = x ⊓ (y ⊓ z ⊔ y \ z) := inf_sup_left.symm
_ = x ⊓ y := by rw [sup_inf_sdiff])
(calc
x ⊓ y ⊓ z ⊓ (x ⊓ y \ z) = x ⊓ x ⊓ (y ⊓ z ⊓ y \ z) := by ac... | Mathlib.Order.BooleanAlgebra.444_0.ewE75DLNneOU8G5 | theorem inf_sdiff_assoc : (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 \ 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_comm _ _ x, inf_comm, inf_sdiff_assoc] | theorem inf_sdiff_right_comm : x \ z ⊓ y = (x ⊓ y) \ z := by
| Mathlib.Order.BooleanAlgebra.455_0.ewE75DLNneOU8G5 | theorem inf_sdiff_right_comm : x \ z ⊓ y = (x ⊓ y) \ z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
a b c : α
⊢ a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c) | /-
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, sdiff_eq_bot_iff.2 inf_le_left, bot_sup_eq, inf_sdiff_assoc] | theorem inf_sdiff_distrib_left (a b c : α) : a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c) := by
| Mathlib.Order.BooleanAlgebra.459_0.ewE75DLNneOU8G5 | theorem inf_sdiff_distrib_left (a b c : α) : a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
a b c : α
⊢ a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) | /-
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... | simp_rw [@inf_comm _ _ _ c, inf_sdiff_distrib_left] | theorem inf_sdiff_distrib_right (a b c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) := by
| Mathlib.Order.BooleanAlgebra.463_0.ewE75DLNneOU8G5 | theorem inf_sdiff_distrib_right (a b c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ Disjoint (x \ z) y ↔ Disjoint 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... | simp_rw [disjoint_iff, inf_sdiff_right_comm, inf_sdiff_assoc] | theorem disjoint_sdiff_comm : Disjoint (x \ z) y ↔ Disjoint x (y \ z) := by
| Mathlib.Order.BooleanAlgebra.467_0.ewE75DLNneOU8G5 | theorem disjoint_sdiff_comm : Disjoint (x \ z) y ↔ Disjoint x (y \ z) | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ 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_left] | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y :=
Eq.symm <|
calc
x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) := by | Mathlib.Order.BooleanAlgebra.471_0.ewE75DLNneOU8G5 | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) = (x \ y ⊔ x ⊔ y \ x) ⊓ (x \ 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... | ac_rfl | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y :=
Eq.symm <|
calc
x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) := by rw [sup_inf_left]
_ = (x \ y ⊔ x ⊔ y \ x) ⊓ (x \ y ⊔ (y \ x ⊔ y)) := by | Mathlib.Order.BooleanAlgebra.471_0.ewE75DLNneOU8G5 | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y | 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 ⊔ (y \ x ⊔ y)) = (x ⊔ y \ 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 [sup_sdiff_right, sup_sdiff_right] | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y :=
Eq.symm <|
calc
x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) := by rw [sup_inf_left]
_ = (x \ y ⊔ x ⊔ y \ x) ⊓ (x \ y ⊔ (y \ x ⊔ y)) := by ac_rfl
_ = (x ⊔ y \ x) ⊓ (x \ y ⊔ y) := by | Mathlib.Order.BooleanAlgebra.471_0.ewE75DLNneOU8G5 | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (x ⊔ y \ x) ⊓ (x \ 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_self_right, sup_sdiff_self_left, inf_idem] | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y :=
Eq.symm <|
calc
x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) := by rw [sup_inf_left]
_ = (x \ y ⊔ x ⊔ y \ x) ⊓ (x \ y ⊔ (y \ x ⊔ y)) := by ac_rfl
_ = (x ⊔ y \ x) ⊓ (x \ y ⊔ y) := by rw [sup_sdiff_right... | Mathlib.Order.BooleanAlgebra.471_0.ewE75DLNneOU8G5 | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y < z \ x
hxz : 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 [← sup_sdiff_cancel_right hxz] | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z := by
| Mathlib.Order.BooleanAlgebra.480_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y < z \ x
hxz : x ≤ z
⊢ x ⊔ 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... | refine' (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le _ | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z := by
rw [← sup_sdiff_cancel_right hxz]
| Mathlib.Order.BooleanAlgebra.480_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y < z \ x
hxz : x ≤ z
h' : x ⊔ 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 [← sdiff_idem] | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z := by
rw [← sup_sdiff_cancel_right hxz]
refine' (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le _
| Mathlib.Order.BooleanAlgebra.480_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : y < z \ x
hxz : x ≤ z
h' : x ⊔ z \ x ≤ x ⊔ y
⊢ (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... | exact (sdiff_le_sdiff_of_sup_le_sup_left h').trans sdiff_le | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z := by
rw [← sup_sdiff_cancel_right hxz]
refine' (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le _
rw [← sdiff_idem]
| Mathlib.Order.BooleanAlgebra.480_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x < z \ y
hyz : 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_sup_cancel hyz] | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z := by
| Mathlib.Order.BooleanAlgebra.487_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x < z \ y
hyz : y ≤ z
⊢ x ⊔ y < 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... | refine' (sup_le_sup_right h.le _).lt_of_not_le fun h' => h.not_le _ | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z := by
rw [← sdiff_sup_cancel hyz]
| Mathlib.Order.BooleanAlgebra.487_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x < z \ y
hyz : y ≤ z
h' : z \ y ⊔ 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 [← sdiff_idem] | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z := by
rw [← sdiff_sup_cancel hyz]
refine' (sup_le_sup_right h.le _).lt_of_not_le fun h' => h.not_le _
| Mathlib.Order.BooleanAlgebra.487_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
h : x < z \ y
hyz : y ≤ z
h' : z \ y ⊔ y ≤ x ⊔ 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... | exact (sdiff_le_sdiff_of_sup_le_sup_right h').trans sdiff_le | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z := by
rw [← sdiff_sup_cancel hyz]
refine' (sup_le_sup_right h.le _).lt_of_not_le fun h' => h.not_le _
rw [← sdiff_idem]
| Mathlib.Order.BooleanAlgebra.487_0.ewE75DLNneOU8G5 | theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : OrderTop α
src✝² : GeneralizedBooleanAlgebra α := inst✝¹
src✝¹ : OrderBot α := toOrderBot
src✝ : OrderTop α := inst✝
x✝¹ x✝ : α
⊢ x✝¹ \ x✝ = x✝¹ ⊓ x✝ᶜ | /-
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... | erw [← inf_sdiff_assoc, inf_top_eq] | /-- A bounded generalized boolean algebra is a boolean algebra. -/
@[reducible]
def GeneralizedBooleanAlgebra.toBooleanAlgebra [GeneralizedBooleanAlgebra α] [OrderTop α] :
BooleanAlgebra α :=
{ ‹GeneralizedBooleanAlgebra α›, GeneralizedBooleanAlgebra.toOrderBot, ‹OrderTop α› with
compl := fun a => ⊤ \ a,
... | Mathlib.Order.BooleanAlgebra.551_0.ewE75DLNneOU8G5 | /-- A bounded generalized boolean algebra is a boolean algebra. -/
@[reducible]
def GeneralizedBooleanAlgebra.toBooleanAlgebra [GeneralizedBooleanAlgebra α] [OrderTop α] :
BooleanAlgebra α | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
src✝ : BooleanAlgebra α := inst✝
a b : α
⊢ a ⊓ b ⊔ a \ b = a | /-
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, ← inf_sup_left, sup_compl_eq_top, inf_top_eq] | instance (priority := 100) BooleanAlgebra.toGeneralizedBooleanAlgebra :
GeneralizedBooleanAlgebra α :=
{ ‹BooleanAlgebra α› with
sup_inf_sdiff := fun a b => by | Mathlib.Order.BooleanAlgebra.600_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
src✝ : BooleanAlgebra α := inst✝
a b : α
⊢ a ⊓ b ⊓ 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... | rw [sdiff_eq, ← inf_inf_distrib_left, inf_compl_eq_bot', inf_bot_eq] | instance (priority := 100) BooleanAlgebra.toGeneralizedBooleanAlgebra :
GeneralizedBooleanAlgebra α :=
{ ‹BooleanAlgebra α› with
sup_inf_sdiff := fun a b => by rw [sdiff_eq, ← inf_sup_left, sup_compl_eq_top, inf_top_eq],
inf_inf_sdiff := fun a b => by
| Mathlib.Order.BooleanAlgebra.600_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
src✝¹ : BooleanAlgebra α := inst✝
src✝ : GeneralizedCoheytingAlgebra α := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra
a b c : α
⊢ a ≤ b ⇨ c ↔ a ⊓ b ≤ c | /-
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 [himp_eq, isCompl_compl.le_sup_right_iff_inf_left_le] | instance (priority := 100) BooleanAlgebra.toBiheytingAlgebra : BiheytingAlgebra α :=
{ ‹BooleanAlgebra α›, GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra with
hnot := compl,
le_himp_iff := fun a b c => by | Mathlib.Order.BooleanAlgebra.609_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
src✝¹ : BooleanAlgebra α := inst✝
src✝ : GeneralizedCoheytingAlgebra α := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra
a : α
⊢ ⊤ \ a = ￢a | /-
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, top_inf_eq] | instance (priority := 100) BooleanAlgebra.toBiheytingAlgebra : BiheytingAlgebra α :=
{ ‹BooleanAlgebra α›, GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra with
hnot := compl,
le_himp_iff := fun a b c => by rw [himp_eq, isCompl_compl.le_sup_right_iff_inf_left_le],
himp_bot := fun _ => _root_.himp_e... | Mathlib.Order.BooleanAlgebra.609_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
src✝¹ : BooleanAlgebra α := inst✝
src✝ : GeneralizedCoheytingAlgebra α := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra
a : α
⊢ aᶜ = ￢a | /-
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... | rfl | instance (priority := 100) BooleanAlgebra.toBiheytingAlgebra : BiheytingAlgebra α :=
{ ‹BooleanAlgebra α›, GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra with
hnot := compl,
le_himp_iff := fun a b c => by rw [himp_eq, isCompl_compl.le_sup_right_iff_inf_left_le],
himp_bot := fun _ => _root_.himp_e... | Mathlib.Order.BooleanAlgebra.609_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : x = yᶜ
⊢ IsCompl 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 [h] | theorem eq_compl_iff_isCompl : x = yᶜ ↔ IsCompl x y :=
⟨fun h => by
| Mathlib.Order.BooleanAlgebra.627_0.ewE75DLNneOU8G5 | theorem eq_compl_iff_isCompl : x = yᶜ ↔ IsCompl x y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : x = yᶜ
⊢ IsCompl yᶜ 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... | exact isCompl_compl.symm | theorem eq_compl_iff_isCompl : x = yᶜ ↔ IsCompl x y :=
⟨fun h => by
rw [h]
| Mathlib.Order.BooleanAlgebra.627_0.ewE75DLNneOU8G5 | theorem eq_compl_iff_isCompl : x = yᶜ ↔ IsCompl x y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : xᶜ = y
⊢ IsCompl 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 [← h] | theorem compl_eq_iff_isCompl : xᶜ = y ↔ IsCompl x y :=
⟨fun h => by
| Mathlib.Order.BooleanAlgebra.633_0.ewE75DLNneOU8G5 | theorem compl_eq_iff_isCompl : xᶜ = y ↔ IsCompl x y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : xᶜ = y
⊢ IsCompl x xᶜ | /-
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 isCompl_compl | theorem compl_eq_iff_isCompl : xᶜ = y ↔ IsCompl x y :=
⟨fun h => by
rw [← h]
| Mathlib.Order.BooleanAlgebra.633_0.ewE75DLNneOU8G5 | theorem compl_eq_iff_isCompl : xᶜ = y ↔ IsCompl x y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ xᶜ = y ↔ yᶜ = 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 [eq_comm, compl_eq_iff_isCompl, eq_compl_iff_isCompl] | theorem compl_eq_comm : xᶜ = y ↔ yᶜ = x := by
| Mathlib.Order.BooleanAlgebra.639_0.ewE75DLNneOU8G5 | theorem compl_eq_comm : xᶜ = y ↔ yᶜ = x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ x = yᶜ ↔ y = xᶜ | /-
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 [eq_comm, compl_eq_iff_isCompl, eq_compl_iff_isCompl] | theorem eq_compl_comm : x = yᶜ ↔ y = xᶜ := by
| Mathlib.Order.BooleanAlgebra.643_0.ewE75DLNneOU8G5 | theorem eq_compl_comm : x = yᶜ ↔ y = xᶜ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : yᶜ ≤ xᶜ
⊢ 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... | have h := compl_le_compl h | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y :=
⟨fun h => by | Mathlib.Order.BooleanAlgebra.697_0.ewE75DLNneOU8G5 | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h✝ : yᶜ ≤ xᶜ
h : xᶜᶜ ≤ yᶜᶜ
⊢ 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... | simp at h | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y :=
⟨fun h => by have h := compl_le_compl h; | Mathlib.Order.BooleanAlgebra.697_0.ewE75DLNneOU8G5 | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h✝ : yᶜ ≤ xᶜ
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... | assumption | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y :=
⟨fun h => by have h := compl_le_compl h; simp at h; | Mathlib.Order.BooleanAlgebra.697_0.ewE75DLNneOU8G5 | @[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
h : yᶜ ≤ x
⊢ xᶜ ≤ 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... | simpa only [compl_compl] using compl_le_compl h | theorem compl_le_of_compl_le (h : yᶜ ≤ x) : xᶜ ≤ y := by
| Mathlib.Order.BooleanAlgebra.705_0.ewE75DLNneOU8G5 | theorem compl_le_of_compl_le (h : yᶜ ≤ x) : xᶜ ≤ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ xᶜ ≤ 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... | simpa using le_compl_self (a := xᶜ) | @[simp] theorem compl_le_self : xᶜ ≤ x ↔ x = ⊤ := by | Mathlib.Order.BooleanAlgebra.713_0.ewE75DLNneOU8G5 | @[simp] theorem compl_le_self : xᶜ ≤ x ↔ x = ⊤ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝¹ : BooleanAlgebra α
inst✝ : Nontrivial α
⊢ xᶜ < 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... | simpa using lt_compl_self (a := xᶜ) | @[simp] theorem compl_lt_self [Nontrivial α] : xᶜ < x ↔ x = ⊤ := by
| Mathlib.Order.BooleanAlgebra.715_0.ewE75DLNneOU8G5 | @[simp] theorem compl_lt_self [Nontrivial α] : xᶜ < x ↔ x = ⊤ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ x \ yᶜ = 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, compl_compl] | @[simp]
theorem sdiff_compl : x \ yᶜ = x ⊓ y := by | Mathlib.Order.BooleanAlgebra.718_0.ewE75DLNneOU8G5 | @[simp]
theorem sdiff_compl : x \ yᶜ = x ⊓ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ x ⊓ y ⊔ x ⊓ yᶜ = 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_eq, sup_inf_sdiff _ _] | @[simp]
theorem sup_inf_inf_compl : x ⊓ y ⊔ x ⊓ yᶜ = x := by | Mathlib.Order.BooleanAlgebra.732_0.ewE75DLNneOU8G5 | @[simp]
theorem sup_inf_inf_compl : x ⊓ y ⊔ x ⊓ yᶜ = x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ (x \ y)ᶜ = 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, himp_eq, compl_inf, compl_compl, sup_comm] | @[simp]
theorem compl_sdiff : (x \ y)ᶜ = x ⇨ y := by
| Mathlib.Order.BooleanAlgebra.736_0.ewE75DLNneOU8G5 | @[simp]
theorem compl_sdiff : (x \ y)ᶜ = x ⇨ y | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ xᶜ \ yᶜ = 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_compl, sdiff_eq, inf_comm] | theorem compl_sdiff_compl : xᶜ \ yᶜ = y \ x := by | Mathlib.Order.BooleanAlgebra.746_0.ewE75DLNneOU8G5 | theorem compl_sdiff_compl : xᶜ \ yᶜ = y \ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ Disjoint xᶜ 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 [← le_compl_iff_disjoint_left, compl_compl] | theorem disjoint_compl_left_iff : Disjoint xᶜ y ↔ y ≤ x := by
| Mathlib.Order.BooleanAlgebra.754_0.ewE75DLNneOU8G5 | theorem disjoint_compl_left_iff : Disjoint xᶜ y ↔ y ≤ x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ Disjoint x yᶜ ↔ 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 [← le_compl_iff_disjoint_right, compl_compl] | theorem disjoint_compl_right_iff : Disjoint x yᶜ ↔ x ≤ y := by
| Mathlib.Order.BooleanAlgebra.758_0.ewE75DLNneOU8G5 | theorem disjoint_compl_right_iff : Disjoint x yᶜ ↔ x ≤ y | Mathlib_Order_BooleanAlgebra |
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ x ⊓ xᶜ ≤ ⊥ | /-
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... | constructor | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case left
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ (x ⊓ xᶜ).1 ≤ ⊥.1 | /-
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... | simp | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by constructor <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case right
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ (x ⊓ xᶜ).2 ≤ ⊥.2 | /-
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... | simp | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by constructor <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ ⊤ ≤ x ⊔ xᶜ | /-
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... | constructor | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by constructor <;> simp
top_le_sup_com... | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case left
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ ⊤.1 ≤ (x ⊔ xᶜ).1 | /-
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... | simp | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by constructor <;> simp
top_le_sup_com... | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case right
α✝ : Type u
β✝ : Type u_1
w x✝ y z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x : α × β
⊢ ⊤.2 ≤ (x ⊔ xᶜ).2 | /-
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... | simp | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> simp [sdiff_eq]
inf_compl_le_bot x := by constructor <;> simp
top_le_sup_com... | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ x \ y = x ⊓ yᶜ | /-
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... | ext | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case a
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ (x \ y).1 = (x ⊓ yᶜ).1 | /-
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... | simp [sdiff_eq] | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case a
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ (x \ y).2 = (x ⊓ yᶜ).2 | /-
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... | simp [sdiff_eq] | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> simp [himp_eq]
sdiff_eq x y := by ext <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ x ⇨ y = y ⊔ xᶜ | /-
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... | ext | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case a
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ (x ⇨ y).1 = (y ⊔ xᶜ).1 | /-
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... | simp [himp_eq] | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
case a
α✝ : Type u
β✝ : Type u_1
w x✝ y✝ z : α✝
α : Type ?u.69761
β : Type ?u.69764
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
src✝¹ : HeytingAlgebra (α × β) := heytingAlgebra
src✝ : DistribLattice (α × β) := distribLattice α β
x y : α × β
⊢ (x ⇨ y).2 = (y ⊔ xᶜ).2 | /-
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... | simp [himp_eq] | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ := Prod.heytingAlgebra
__ := Prod.distribLattice α β
himp_eq x y := by ext <;> | Mathlib.Order.BooleanAlgebra.784_0.ewE75DLNneOU8G5 | instance Prod.booleanAlgebra (α β) [BooleanAlgebra α] [BooleanAlgebra β] :
BooleanAlgebra (α × β) where
__ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁴ : Sup α
inst✝³ : Inf α
inst✝² : Bot α
inst✝¹ : SDiff α
inst✝ : GeneralizedBooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_bot : f ⊥ = ⊥
map_sdiff : ∀ (a b : α), f (a \ b) = f a \ f b
src✝... | /-
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... | erw [map_sup, map_sdiff, map_inf, sup_inf_sdiff] | /-- Pullback a `GeneralizedBooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.generalizedBooleanAlgebra [Sup α] [Inf α] [Bot α] [SDiff α]
[GeneralizedBooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f ... | Mathlib.Order.BooleanAlgebra.828_0.ewE75DLNneOU8G5 | /-- Pullback a `GeneralizedBooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.generalizedBooleanAlgebra [Sup α] [Inf α] [Bot α] [SDiff α]
[GeneralizedBooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f ... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁴ : Sup α
inst✝³ : Inf α
inst✝² : Bot α
inst✝¹ : SDiff α
inst✝ : GeneralizedBooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_bot : f ⊥ = ⊥
map_sdiff : ∀ (a b : α), f (a \ b) = f a \ f b
src✝... | /-
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... | erw [map_inf, map_sdiff, map_inf, inf_inf_sdiff, map_bot] | /-- Pullback a `GeneralizedBooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.generalizedBooleanAlgebra [Sup α] [Inf α] [Bot α] [SDiff α]
[GeneralizedBooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f ... | Mathlib.Order.BooleanAlgebra.828_0.ewE75DLNneOU8G5 | /-- Pullback a `GeneralizedBooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.generalizedBooleanAlgebra [Sup α] [Inf α] [Bot α] [SDiff α]
[GeneralizedBooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f ... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁶ : Sup α
inst✝⁵ : Inf α
inst✝⁴ : Top α
inst✝³ : Bot α
inst✝² : HasCompl α
inst✝¹ : SDiff α
inst✝ : BooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_top : f ⊤ = ⊤
map_bot : f ⊥ = ⊥
map_compl... | /-
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 [map_compl, inf_compl_eq_bot, map_bot] | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib.Order.BooleanAlgebra.842_0.ewE75DLNneOU8G5 | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁶ : Sup α
inst✝⁵ : Inf α
inst✝⁴ : Top α
inst✝³ : Bot α
inst✝² : HasCompl α
inst✝¹ : SDiff α
inst✝ : BooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_top : f ⊤ = ⊤
map_bot : f ⊥ = ⊥
map_compl... | /-
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 [map_compl, sup_compl_eq_top, map_top] | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib.Order.BooleanAlgebra.842_0.ewE75DLNneOU8G5 | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁶ : Sup α
inst✝⁵ : Inf α
inst✝⁴ : Top α
inst✝³ : Bot α
inst✝² : HasCompl α
inst✝¹ : SDiff α
inst✝ : BooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_top : f ⊤ = ⊤
map_bot : f ⊥ = ⊥
map_compl... | /-
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 hf ((map_sdiff _ _).trans (sdiff_eq.trans ?_)) | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib.Order.BooleanAlgebra.842_0.ewE75DLNneOU8G5 | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝⁶ : Sup α
inst✝⁵ : Inf α
inst✝⁴ : Top α
inst✝³ : Bot α
inst✝² : HasCompl α
inst✝¹ : SDiff α
inst✝ : BooleanAlgebra β
f : α → β
hf : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
map_top : f ⊤ = ⊤
map_bot : f ⊥ = ⊥
map_compl... | /-
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 [map_inf, map_compl] | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib.Order.BooleanAlgebra.842_0.ewE75DLNneOU8G5 | /-- Pullback a `BooleanAlgebra` along an injection. -/
@[reducible]
protected def Function.Injective.booleanAlgebra [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α]
[SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
⊢ BooleanAlgebra PUnit.{?u.90848 + 1} | /-
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'
{ PUnit.biheytingAlgebra with
.. } | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
| Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_1
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
⊢ ∀ (x y z : PUnit.{?u.90854 + 1}), (x ⊔ y) ⊓ (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... | intros | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> ( | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_1
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
x✝ y✝ z✝ : PUnit.{?u.90854 + 1}
⊢ (x✝ ⊔ y✝) ⊓ (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... | trivial | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> (intros; | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_2
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
⊢ ∀ (x : PUnit.{?u.90854 + 1}), x ⊓ xᶜ ≤ ⊥ | /-
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... | intros | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> ( | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_2
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
x✝ : PUnit.{?u.90854 + 1}
⊢ x✝ ⊓ x✝ᶜ ≤ ⊥ | /-
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... | trivial | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> (intros; | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_3
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
⊢ ∀ (x : PUnit.{?u.90854 + 1}), ⊤ ≤ x ⊔ xᶜ | /-
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... | intros | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> ( | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_3
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
x✝ : PUnit.{?u.90854 + 1}
⊢ ⊤ ≤ x✝ ⊔ x✝ᶜ | /-
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... | trivial | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> (intros; | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_4
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
⊢ ∀ (x y : PUnit.{?u.90854 + 1}), x \ y = x ⊓ yᶜ | /-
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... | intros | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> ( | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_4
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
x✝ y✝ : PUnit.{?u.90854 + 1}
⊢ x✝ \ y✝ = x✝ ⊓ y✝ᶜ | /-
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... | trivial | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> (intros; | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_5
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
⊢ ∀ (x y : PUnit.{?u.90854 + 1}), x ⇨ y = y ⊔ xᶜ | /-
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... | intros | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> ( | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
case refine'_5
α : Type u
β : Type u_1
w x y z : α
src✝ : BiheytingAlgebra PUnit.{?u.90854 + 1} := biheytingAlgebra
x✝ y✝ : PUnit.{?u.90854 + 1}
⊢ x✝ ⇨ y✝ = y✝ ⊔ x✝ᶜ | /-
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... | trivial | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit := by
refine'
{ PUnit.biheytingAlgebra with
.. } <;> (intros; | Mathlib.Order.BooleanAlgebra.865_0.ewE75DLNneOU8G5 | instance PUnit.booleanAlgebra : BooleanAlgebra PUnit | Mathlib_Order_BooleanAlgebra |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
nth_s_eq : Stream'.Seq.get? g.s n = some gp
nth_conts_aux_eq : continuantsAux g n = ppred
succ_nth_conts_aux_eq : continuantsAux g (n + 1) = pred
⊢ continuantsAux g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := g... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | simp [*, continuantsAux, nextContinuants, nextDenominator, nextNumerator] | theorem continuantsAux_recurrence {gp ppred pred : Pair K} (nth_s_eq : g.s.get? n = some gp)
(nth_conts_aux_eq : g.continuantsAux n = ppred)
(succ_nth_conts_aux_eq : g.continuantsAux (n + 1) = pred) :
g.continuantsAux (n + 2) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ :=
by | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.26_0.nOytPSFGrohRR6p | theorem continuantsAux_recurrence {gp ppred pred : Pair K} (nth_s_eq : g.s.get? n = some gp)
(nth_conts_aux_eq : g.continuantsAux n = ppred)
(succ_nth_conts_aux_eq : g.continuantsAux (n + 1) = pred) :
g.continuantsAux (n + 2) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
nth_s_eq : Stream'.Seq.get? g.s n = some gp
nth_conts_aux_eq : continuantsAux g n = ppred
succ_nth_conts_aux_eq : continuantsAux g (n + 1) = pred
⊢ continuants g (n + 1) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | simp [nth_cont_eq_succ_nth_cont_aux,
continuantsAux_recurrence nth_s_eq nth_conts_aux_eq succ_nth_conts_aux_eq] | theorem continuants_recurrenceAux {gp ppred pred : Pair K} (nth_s_eq : g.s.get? n = some gp)
(nth_conts_aux_eq : g.continuantsAux n = ppred)
(succ_nth_conts_aux_eq : g.continuantsAux (n + 1) = pred) :
g.continuants (n + 1) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ := by
| Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.33_0.nOytPSFGrohRR6p | theorem continuants_recurrenceAux {gp ppred pred : Pair K} (nth_s_eq : g.s.get? n = some gp)
(nth_conts_aux_eq : g.continuantsAux n = ppred)
(succ_nth_conts_aux_eq : g.continuantsAux (n + 1) = pred) :
g.continuants (n + 1) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_conts_eq : continuants g n = ppred
succ_nth_conts_eq : continuants g (n + 1) = pred
⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * ... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | rw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂` and `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem continuants_recurrence {gp ppred pred : Pair K} (succ_nth_s_eq : g.s.get? (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred) (succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.41_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂` and `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem continuants_recurrence {gp ppred pred : Pair K} (succ_nth_s_eq : g.s.get? (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred) (succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_conts_eq : continuantsAux g (n + 1) = ppred
succ_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred
⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | exact continuants_recurrenceAux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂` and `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem continuants_recurrence {gp ppred pred : Pair K} (succ_nth_s_eq : g.s.get? (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred) (succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.41_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂` and `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem continuants_recurrence {gp ppred pred : Pair K} (succ_nth_s_eq : g.s.get? (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred) (succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
ppredA predA : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_num_eq : numerators g n = ppredA
succ_nth_num_eq : numerators g (n + 1) = predA
⊢ numerators g (n + 2) = gp.b * predA + gp.a * ppredA | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.a = ppredA | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by
| Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.49_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
ppredA predA : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_num_eq : numerators g n = ppredA
succ_nth_num_eq : numerators g (n + 1) = predA
⊢ ∃ conts, continuants g n = conts ∧ conts.a = ppredA
case intro.intro.refl... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | exact exists_conts_a_of_num nth_num_eq | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by
ob... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.49_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
case intro.intro.refl
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
predA : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
succ_nth_num_eq : numerators g (n + 1) = predA
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_num_eq : numerators g n = ppr... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.a = predA | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by
ob... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.49_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
predA : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
succ_nth_num_eq : numerators g (n + 1) = predA
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_num_eq : numerators g n = ppredConts.a
⊢ ∃ conts, c... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | exact exists_conts_a_of_num succ_nth_num_eq | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by
ob... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.49_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
case intro.intro.refl.intro.intro.refl
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_num_eq : numerators g n = ppredConts.a
predConts : Pair K
succ_nth_co... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | rw [num_eq_conts_a, continuants_recurrence succ_nth_s_eq nth_conts_eq succ_nth_conts_eq] | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by
ob... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.49_0.nOytPSFGrohRR6p | /-- Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. -/
theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
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