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 v
α : Type u
inst✝⁴ : Lattice α
inst✝³ : DecidableEq α
inst✝² : DecidableRel fun x x_1 => x ≤ x_1
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
src✝ : Lattice α := inst✝⁴
⊢ ∀ (a b : α), min a b = if a ≤ b then a else b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact congr_fun₂ inf_eq_minDefault | /-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
... | Mathlib.Order.Lattice.909_0.wE3igZl9MFbJBfv | /-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α | Mathlib_Order_Lattice |
α✝ : Type u
β : Type v
α : Type u
inst✝⁴ : Lattice α
inst✝³ : DecidableEq α
inst✝² : DecidableRel fun x x_1 => x ≤ x_1
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
src✝ : Lattice α := inst✝⁴
⊢ ∀ (a b : α), max a b = if a ≤ b then b else a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact congr_fun₂ sup_eq_maxDefault | /-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
... | Mathlib.Order.Lattice.909_0.wE3igZl9MFbJBfv | /-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α | Mathlib_Order_Lattice |
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
⊢ update f i (a ⊔ b) j = (update f i a ⊔ update f i b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | obtain rfl | hji := eq_or_ne j i | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by | Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b | Mathlib_Order_Lattice |
case inl
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
j : ι
a b : π j
⊢ update f j (a ⊔ b) j = (update f j a ⊔ update f j b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp [update_noteq, *] | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> | Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b | Mathlib_Order_Lattice |
case inr
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
hji : j ≠ i
⊢ update f i (a ⊔ b) j = (update f i a ⊔ update f i b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp [update_noteq, *] | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> | Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv | theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b | Mathlib_Order_Lattice |
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
⊢ update f i (a ⊓ b) j = (update f i a ⊓ update f i b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | obtain rfl | hji := eq_or_ne j i | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by | Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b | Mathlib_Order_Lattice |
case inl
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
j : ι
a b : π j
⊢ update f j (a ⊓ b) j = (update f j a ⊓ update f j b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp [update_noteq, *] | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> | Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b | Mathlib_Order_Lattice |
case inr
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
hji : j ≠ i
⊢ update f i (a ⊓ b) j = (update f i a ⊓ update f i b) j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp [update_noteq, *] | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> | Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv | theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
f : α → β
h : ∀ (x y : α), f (x ⊓ y) = f x ⊓ f y
x y : α
hxy : x ≤ y
⊢ f x ⊓ f y = f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [← h, inf_eq_left.2 hxy] | theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by | Mathlib.Order.Lattice.1099_0.wE3igZl9MFbJBfv | theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [h, hf h, sup_of_le_right] | theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by | Mathlib.Order.Lattice.1111_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
f : α → β
hf : Monotone f
x y : α
h : y ≤ x
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [h, hf h, sup_of_le_left] | theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
| Mathlib.Order.Lattice.1111_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
α : Type u
β : Type v
f : α → β
s : Set α
x✝ y✝ : α
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊓ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f x ⊓ f y = f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [← h _ hx _ hy, inf_eq_left.2 hxy] | theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by | Mathlib.Order.Lattice.1151_0.wE3igZl9MFbJBfv | theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s | Mathlib_Order_Lattice |
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | cases le_total x y | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | have := hf ?_ ?_ ‹_› | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | have := hf ?_ ?_ ‹_› | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y | Mathlib_Order_Lattice |
α : Type u
β : Type v
f : α → β
s : Set α
x✝ y✝ : α
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeSup β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊔ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f x ⊔ f y = f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [← h _ hx _ hy, inf_eq_left.2 hxy] | theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by | Mathlib.Order.Lattice.1257_0.wE3igZl9MFbJBfv | theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s | Mathlib_Order_Lattice |
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | cases le_total x y | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | have := hf ?_ ?_ ‹_› | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | have := hf ?_ ?_ ‹_› | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
| Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | assumption | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| | Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv | theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ a ≤ 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f a ≤ f (a ⊔ b) | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ≤ f (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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_sup] | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ≤ f a ⊔ f b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact le_sup_left | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f b ≤ f (a ⊔ b) | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f b ≤ f (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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_sup] | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f b ≤ f a ⊔ f b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact le_sup_right | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f (a ⊔ b) ≤ f c | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : b ≤ c
⊢ f (a ⊔ b) ≤ f c | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_sup] | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : b ≤ c
⊢ f a ⊔ f b ≤ f c | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact sup_le ha hb | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f (a ⊓ b) ≤ f a | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f (a ⊓ b) ≤ f a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_inf] | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ⊓ f b ≤ f a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact inf_le_left | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ a ⊓ b ≤ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f (a ⊓ b) ≤ f b | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f (a ⊓ b) ≤ f b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_inf] | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ⊓ f b ≤ f b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact inf_le_right | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f a ≤ f (b ⊓ c) | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ c
⊢ f a ≤ f (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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_inf] | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ c
⊢ f a ≤ f b ⊓ f c | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact le_inf ha hb | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : 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
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ (a ⊔ b) ⊓ (a ⊔ 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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c) | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : 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
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (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
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | rw [map_inf, map_sup, map_sup, map_sup, map_inf] | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib_Order_Lattice |
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : 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
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ (f a ⊔ f b) ⊓ (f a ⊔ f c) ≤ f a ⊔ f b ⊓ f c | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | exact le_sup_inf | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv | /-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f... | Mathlib_Order_Lattice |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p : MvPolynomial σ R
⊢ (map f) ((rename g) p) = (rename g) ((map f) p) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul] | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
| Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p : MvPolynomial σ R
a : R
⊢ (map f) ((rename g) (C a)) = (rename g) ((map f) (C a)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [map_C, rename_C] | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by | Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p✝ p q : MvPolynomial σ R
hp : (map f) ((rename g) p) = (rename g) ((map f) p)
hq : (map f) ((rename g) q) = (rename g) ((map f) q)
⊢ (map f) ((rename g) (p + q)) = (rename g) ((map f) (... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [hp, hq, AlgHom.map_add, RingHom.map_add] | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by | Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p✝ p : MvPolynomial σ R
n : σ
hp : (map f) ((rename g) p) = (rename g) ((map f) p)
⊢ (map f) ((rename g) (p * X n)) = (rename g) ((map f) (p * X n)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul] | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
| Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ (rename g) (eval₂ C (X ∘ f) p) = (rename (g ∘ f)) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [rename, aeval_eq_eval₂Hom] | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
| Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) (eval₂ C (X ∘ f) p) =
(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p] | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hop... | Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C)
(⇑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) ∘ X ∘ f) p =
(eval₂Hom (... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [(· ∘ ·), eval₂Hom_X'] | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hop... | Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) (fun x => X (g (f x))) p =
(eval₂Hom (algebraMap R (MvPolynomial α R)) fun x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' eval₂Hom_congr _ rfl rfl | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hop... | Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C = algebraMap R (MvPolynomial α R) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | ext1 | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hop... | Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
case a
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
x✝ : R
⊢ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) x✝ =
(algebraMap R (MvPolynomial α R)) x✝ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C] | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hop... | Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK | @[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ (rename f) ((monomial d) r) = (monomial (Finsupp.mapDomain f d)) r | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index] | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
| Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ((algebraMap R (MvPolynomial τ R)) r * Finsupp.prod d fun i k => (X ∘ f) i ^ k) =
C r * Finsupp.prod d fun a m => X (f a) ^ m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· | Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r | Mathlib_Data_MvPolynomial_Rename |
case h_zero
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ∀ (b : τ), X b ^ 0 = 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | exact fun n => pow_zero _ | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· | Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ∀ (b : τ) (m₁ m₂ : ℕ), X b ^ (m₁ + m₂) = X b ^ m₁ * X b ^ m₂ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | exact fun n i₁ i₂ => pow_add _ _ _ | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· | Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
p : MvPolynomial σ R
⊢ (rename f) p = Finsupp.mapDomain (Finsupp.mapDomain f) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index] | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
| Mathlib.Data.MvPolynomial.Rename.107_0.3NqVCwOs1E93kvK | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
p : MvPolynomial σ R
⊢ (sum p fun s a => (monomial (sum s fun a b => fun₀ | f a => b)) a) =
sum p fun a => Finsupp.single (sum a fun a => Finsupp.single (f a)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
| Mathlib.Data.MvPolynomial.Rename.107_0.3NqVCwOs1E93kvK | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
⊢ Injective ⇑(rename f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f) | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
| Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
⊢ Injective ⇑(rename f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [this] | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
| Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
⊢ Injective (Finsupp.mapDomain (Finsupp.mapDomain f)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf) | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
| Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
i : σ
⊢ (AlgHom.comp (killCompl hf) (rename f)) (X i) = (AlgHom.id R (MvPolynomial σ R)) (X i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | dsimp | theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
| Mathlib.Data.MvPolynomial.Rename.136_0.3NqVCwOs1E93kvK | theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
i : σ
⊢ (killCompl hf) ((rename f) (X i)) = X i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply] | theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
| Mathlib.Data.MvPolynomial.Rename.136_0.3NqVCwOs1E93kvK | theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ ≃ τ
src✝ : MvPolynomial σ R →ₐ[R] MvPolynomial τ R := rename ⇑f
p : MvPolynomial σ R
⊢ (rename ⇑f.symm) ((rename ⇑f) p) = p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_rename, f.symm_comp_self, rename_id] | /-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by | Mathlib.Data.MvPolynomial.Rename.153_0.3NqVCwOs1E93kvK | /-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ ≃ τ
src✝ : MvPolynomial σ R →ₐ[R] MvPolynomial τ R := rename ⇑f
p : MvPolynomial τ R
⊢ (rename ⇑f) ((rename ⇑f.symm) p) = p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_rename, f.self_comp_symm, rename_id] | /-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => ... | Mathlib.Data.MvPolynomial.Rename.153_0.3NqVCwOs1E93kvK | /-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
| Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (a : R), eval₂ f g ((rename k) (C a)) = eval₂ f (g ∘ k) (C a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
a✝ : R
⊢ eval₂ f g ((rename k) (C a✝)) = eval₂ f (g ∘ k) (C a✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p →
eval₂ f g ((rename k) q) = eval₂ f (g ∘ k) q → eval₂ f g ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p p✝ q✝ : MvPolynomial σ R
a✝¹ : eval₂ f g ((rename k) p✝) = eval₂ f (g ∘ k) p✝
a✝ : eval₂ f g ((rename k) q✝) = eval₂ f (g ∘ k) q✝
⊢ eval₂ f g ((rename k) (p✝ + q✝)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p → eval₂ f g ((rename k) (p * X n)) = eval₂ f (g ∘ k) (p * X n... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p p✝ : MvPolynomial σ R
n✝ : σ
a✝ : eval₂ f g ((rename k) p✝) = eval₂ f (g ∘ k) p✝
⊢ eval₂ f g ((rename k) (p✝ * X n✝)) = eval₂ f (g ∘ k) (p✝ * X n✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK | theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ (rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
| Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ ∀ (a : R), (rename k) (eval₂ C (g ∘ k) (C a)) = eval₂ C (⇑(rename k) ∘ g) ((rename k) (C a)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
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