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 |
|---|---|---|---|---|---|---|
case none
α✝ : Type ?u.229
β✝ : Type ?u.232
γ✝ : Type ?u.235
f✝ : α✝ → β✝ → γ✝
a : Option α✝
b✝ : Option β✝
c : Option γ✝
α β γ : Type u
f : α → β → γ
b : Option β
⊢ map₂ f none b = Seq.seq (f <$> none) fun x => b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b :=
by cases a <;> | Mathlib.Data.Option.NAry.46_0.qAUZsy5CifjOLJ4 | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b | Mathlib_Data_Option_NAry |
case some
α✝ : Type ?u.229
β✝ : Type ?u.232
γ✝ : Type ?u.235
f✝ : α✝ → β✝ → γ✝
a : Option α✝
b✝ : Option β✝
c : Option γ✝
α β γ : Type u
f : α → β → γ
b : Option β
val✝ : α
⊢ map₂ f (some val✝) b = Seq.seq (f <$> some val✝) fun x => b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b :=
by cases a <;> | Mathlib.Data.Option.NAry.46_0.qAUZsy5CifjOLJ4 | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b | Mathlib_Data_Option_NAry |
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a✝ : Option α
b : Option β
c : Option γ
f : α → β → γ
a : Option α
⊢ map₂ f a none = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by | Mathlib.Data.Option.NAry.64_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none | Mathlib_Data_Option_NAry |
case none
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
⊢ map₂ f none none = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by cases a <;> | Mathlib.Data.Option.NAry.64_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none | Mathlib_Data_Option_NAry |
case some
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
val✝ : α
⊢ map₂ f (some val✝) none = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by cases a <;> | Mathlib.Data.Option.NAry.64_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none | Mathlib_Data_Option_NAry |
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a✝ : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
a : Option α
b : β
⊢ map₂ f a (some b) = Option.map (fun a => f a b) a | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b :=
by | Mathlib.Data.Option.NAry.74_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b | Mathlib_Data_Option_NAry |
case none
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
b : β
⊢ map₂ f none (some b) = Option.map (fun a => f a b) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b :=
by cases a <;> | Mathlib.Data.Option.NAry.74_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b | Mathlib_Data_Option_NAry |
case some
α : Type u_1
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
b : β
val✝ : α
⊢ map₂ f (some val✝) (some b) = Option.map (fun a => f a b) (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b :=
by cases a <;> | Mathlib.Data.Option.NAry.74_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) : map₂ f a b = a.map fun a => f a b | Mathlib_Data_Option_NAry |
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
a : Option α
b : Option β
c✝ : Option γ
c : γ
⊢ c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [map₂] | theorem mem_map₂_iff {c : γ} : c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c :=
by | Mathlib.Data.Option.NAry.80_0.qAUZsy5CifjOLJ4 | theorem mem_map₂_iff {c : γ} : c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c | Mathlib_Data_Option_NAry |
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
a : Option α
b : Option β
c : Option γ
⊢ map₂ f a b = none ↔ a = none ∨ b = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
b : Option β
c : Option γ
⊢ map₂ f none b = none ↔ none = none ∨ b = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
b : Option β
c : Option γ
val✝ : α
⊢ map₂ f (some val✝) b = none ↔ some val✝ = none ∨ b = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case none.none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
⊢ map₂ f none none = none ↔ none = none ∨ none = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case none.some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
val✝ : β
⊢ map₂ f none (some val✝) = none ↔ none = none ∨ some val✝ = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case some.none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
val✝ : α
⊢ map₂ f (some val✝) none = none ↔ some val✝ = none ∨ none = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
case some.some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
val✝¹ : α
val✝ : β
⊢ map₂ f (some val✝¹) (some val✝) = none ↔ some val✝¹ = none ∨ some val✝ = none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.84_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none | Mathlib_Data_Option_NAry |
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a✝ : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
a : Option α
b : Option β
⊢ map₂ f a b = map₂ (fun a b => f b a) b a | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case none
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
b : Option β
⊢ map₂ f none b = map₂ (fun a b => f b a) b none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case some
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b✝ : Option β
c : Option γ
f : α → β → γ
b : Option β
val✝ : α
⊢ map₂ f (some val✝) b = map₂ (fun a b => f b a) b (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case none.none
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
⊢ map₂ f none none = map₂ (fun a b => f b a) none none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case none.some
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
val✝ : β
⊢ map₂ f none (some val✝) = map₂ (fun a b => f b a) (some val✝) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case some.none
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
val✝ : α
⊢ map₂ f (some val✝) none = map₂ (fun a b => f b a) none (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
case some.some
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
val✝¹ : α
val✝ : β
⊢ map₂ f (some val✝¹) (some val✝) = map₂ (fun a b => f b a) (some val✝) (some val✝¹) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.89_0.qAUZsy5CifjOLJ4 | theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = map₂ (fun a b => f b a) b a | Mathlib_Data_Option_NAry |
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
⊢ Option.map g (map₂ f a b) = map₂ (fun a b => g (f a b)) a b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case none
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
⊢ Option.map g (map₂ f none b) = map₂ (fun a b => g (f a b)) none b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case some
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
val✝ : α
⊢ Option.map g (map₂ f (some val✝) b) = map₂ (fun a b => g (f a b)) (some val✝) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case none.none
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
⊢ Option.map g (map₂ f none none) = map₂ (fun a b => g (f a b)) none none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case none.some
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
val✝ : β
⊢ Option.map g (map₂ f none (some val✝)) = map₂ (fun a b => g (f a b)) none (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case some.none
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
val✝ : α
⊢ Option.map g (map₂ f (some val✝) none) = map₂ (fun a b => g (f a b)) (some val✝) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
case some.some
α : Type u_3
β : Type u_4
γ : Type u_2
f✝ : α → β → γ
c : Option γ
δ : Type u_1
f : α → β → γ
g : γ → δ
val✝¹ : α
val✝ : β
⊢ Option.map g (map₂ f (some val✝¹) (some val✝)) = map₂ (fun a b => g (f a b)) (some val✝¹) (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.93_0.qAUZsy5CifjOLJ4 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b | Mathlib_Data_Option_NAry |
α : Type u_4
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
f : γ → β → δ
g : α → γ
⊢ map₂ f (Option.map g a) b = map₂ (fun a b => f (g a) b) a b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by | Mathlib.Data.Option.NAry.97_0.qAUZsy5CifjOLJ4 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b | Mathlib_Data_Option_NAry |
case none
α : Type u_4
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
f : γ → β → δ
g : α → γ
⊢ map₂ f (Option.map g none) b = map₂ (fun a b => f (g a) b) none b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by cases a <;> | Mathlib.Data.Option.NAry.97_0.qAUZsy5CifjOLJ4 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b | Mathlib_Data_Option_NAry |
case some
α : Type u_4
β : Type u_3
γ : Type u_2
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
f : γ → β → δ
g : α → γ
val✝ : α
⊢ map₂ f (Option.map g (some val✝)) b = map₂ (fun a b => f (g a) b) (some val✝) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by cases a <;> | Mathlib.Data.Option.NAry.97_0.qAUZsy5CifjOLJ4 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b | Mathlib_Data_Option_NAry |
α : Type u_2
β : Type u_4
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
f : α → γ → δ
g : β → γ
⊢ map₂ f a (Option.map g b) = map₂ (fun a b => f a (g b)) a b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by | Mathlib.Data.Option.NAry.101_0.qAUZsy5CifjOLJ4 | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b | Mathlib_Data_Option_NAry |
case none
α : Type u_2
β : Type u_4
γ : Type u_3
f✝ : α → β → γ
a : Option α
c : Option γ
δ : Type u_1
f : α → γ → δ
g : β → γ
⊢ map₂ f a (Option.map g none) = map₂ (fun a b => f a (g b)) a none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by cases b <;> | Mathlib.Data.Option.NAry.101_0.qAUZsy5CifjOLJ4 | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b | Mathlib_Data_Option_NAry |
case some
α : Type u_2
β : Type u_4
γ : Type u_3
f✝ : α → β → γ
a : Option α
c : Option γ
δ : Type u_1
f : α → γ → δ
g : β → γ
val✝ : β
⊢ map₂ f a (Option.map g (some val✝)) = map₂ (fun a b => f a (g b)) a (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by cases b <;> | Mathlib.Data.Option.NAry.101_0.qAUZsy5CifjOLJ4 | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b | Mathlib_Data_Option_NAry |
α : Type u_2
β : Type u_1
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
x : Option (α × β)
⊢ Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases x | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) := by | Mathlib.Data.Option.NAry.110_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) | Mathlib_Data_Option_NAry |
case none
α : Type u_2
β : Type u_1
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
⊢ Option.map (uncurry f) none = map₂ f (Option.map Prod.fst none) (Option.map Prod.snd none) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) := by cases x <;> | Mathlib.Data.Option.NAry.110_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) | Mathlib_Data_Option_NAry |
case some
α : Type u_2
β : Type u_1
γ : Type u_3
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
f : α → β → γ
val✝ : α × β
⊢ Option.map (uncurry f) (some val✝) = map₂ f (Option.map Prod.fst (some val✝)) (Option.map Prod.snd (some val✝)) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | rfl | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) := by cases x <;> | Mathlib.Data.Option.NAry.110_0.qAUZsy5CifjOLJ4 | @[simp]
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) | Mathlib_Data_Option_NAry |
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
⊢ map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
⊢ map₂ f (map₂ g none b) c = map₂ f' none (map₂ g' b c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : α
⊢ map₂ f (map₂ g (some val✝) b) c = map₂ f' (some val✝) ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
⊢ map₂ f (map₂ g none none) c = map₂ f' none (map₂ g' none c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : β
⊢ map₂ f (map₂ g none (some val✝)) c = map₂ f' none (map₂ g' (so... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : α
⊢ map₂ f (map₂ g (some val✝) none) c = map₂ f' (some val✝) (map₂... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝¹ : α
val✝ : β
⊢ map₂ f (map₂ g (some val✝¹) (some val✝)) c = map₂ f'... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
⊢ map₂ f (map₂ g none none) none = map₂ f' none (map₂ g' none none) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : γ
⊢ map₂ f (map₂ g none none) (some val✝) = map₂ f' none (map₂ g' none (so... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : β
⊢ map₂ f (map₂ g none (some val✝)) none = map₂ f' none (map₂ g' (some va... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case none.some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝¹ : β
val✝ : γ
⊢ map₂ f (map₂ g none (some val✝¹)) (some val✝) = map₂ f' none... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝ : α
⊢ map₂ f (map₂ g (some val✝) none) none = map₂ f' (some val✝) (map₂ g' n... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝¹ : α
val✝ : γ
⊢ map₂ f (map₂ g (some val✝¹) none) (some val✝) = map₂ f' (som... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝¹ : α
val✝ : β
⊢ map₂ f (map₂ g (some val✝¹) (some val✝)) none = map₂ f' (som... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
case some.some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
ε' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
val✝² : α
val✝¹ : β
val✝ : γ
⊢ map₂ f (map₂ g (some val✝²) (some val✝¹)) (some va... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_assoc] | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.124_0.qAUZsy5CifjOLJ4 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) | Mathlib_Data_Option_NAry |
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
a : Option α
b : Option β
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
⊢ map₂ f a b = map₂ g b a | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
b : Option β
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
⊢ map₂ f none b = map₂ g b none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
b : Option β
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
val✝ : α
⊢ map₂ f (some val✝) b = map₂ g b (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case none.none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
⊢ map₂ f none none = map₂ g none none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_comm] | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case none.some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
val✝ : β
⊢ map₂ f none (some val✝) = map₂ g (some val✝) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_comm] | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case some.none
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
val✝ : α
⊢ map₂ f (some val✝) none = map₂ g none (some val✝) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_comm] | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
case some.some
α : Type u_2
β : Type u_3
γ : Type u_1
f : α → β → γ
c : Option γ
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
val✝¹ : α
val✝ : β
⊢ map₂ f (some val✝¹) (some val✝) = map₂ g (some val✝) (some val✝¹) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_comm] | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.130_0.qAUZsy5CifjOLJ4 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a | Mathlib_Data_Option_NAry |
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ map₂ f none (map₂ g b c) = map₂ g' b (map₂ f' none c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : α
⊢ map₂ f (some val✝) (map₂ g b c) = map₂ g' b (map₂ ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ map₂ f none (map₂ g none c) = map₂ g' none (map₂ f' none c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : β
⊢ map₂ f none (map₂ g (some val✝) c) = map₂ g' (some val✝) (... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : α
⊢ map₂ f (some val✝) (map₂ g none c) = map₂ g' none (map₂ f'... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝¹ : α
val✝ : β
⊢ map₂ f (some val✝¹) (map₂ g (some val✝) c) = map... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.none.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ map₂ f none (map₂ g none none) = map₂ g' none (map₂ f' none none) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.none.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : γ
⊢ map₂ f none (map₂ g none (some val✝)) = map₂ g' none (map₂ f' none... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.some.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : β
⊢ map₂ f none (map₂ g (some val✝) none) = map₂ g' (some val✝) (map₂ ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case none.some.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝¹ : β
val✝ : γ
⊢ map₂ f none (map₂ g (some val✝¹) (some val✝)) = map₂ g' ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.none.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝ : α
⊢ map₂ f (some val✝) (map₂ g none none) = map₂ g' none (map₂ f' (som... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.none.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝¹ : α
val✝ : γ
⊢ map₂ f (some val✝¹) (map₂ g none (some val✝)) = map₂ g' ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.some.none
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝¹ : α
val✝ : β
⊢ map₂ f (some val✝¹) (map₂ g (some val✝) none) = map₂ g' ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
case some.some.some
α : Type u_4
β : Type u_5
γ : Type u_6
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
val✝² : α
val✝¹ : β
val✝ : γ
⊢ map₂ f (some val✝²) (map₂ g (some val✝¹) (some... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_left_comm] | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.134_0.qAUZsy5CifjOLJ4 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) | Mathlib_Data_Option_NAry |
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
⊢ map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
⊢ map₂ f (map₂ g none b) c = map₂ g' (map₂ f' none c) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : α
⊢ map₂ f (map₂ g (some val✝) b) c = map₂ g' (map₂ f... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
⊢ map₂ f (map₂ g none none) c = map₂ g' (map₂ f' none c) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : β
⊢ map₂ f (map₂ g none (some val✝)) c = map₂ g' (map₂ f' non... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : α
⊢ map₂ f (map₂ g (some val✝) none) c = map₂ g' (map₂ f' (so... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
c : Option γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝¹ : α
val✝ : β
⊢ map₂ f (map₂ g (some val✝¹) (some val✝)) c = ma... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases c | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
⊢ map₂ f (map₂ g none none) none = map₂ g' (map₂ f' none none) none | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : γ
⊢ map₂ f (map₂ g none none) (some val✝) = map₂ g' (map₂ f' none (so... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : β
⊢ map₂ f (map₂ g none (some val✝)) none = map₂ g' (map₂ f' none non... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case none.some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝¹ : β
val✝ : γ
⊢ map₂ f (map₂ g none (some val✝¹)) (some val✝) = map₂ g'... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝ : α
⊢ map₂ f (map₂ g (some val✝) none) none = map₂ g' (map₂ f' (some va... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝¹ : α
val✝ : γ
⊢ map₂ f (map₂ g (some val✝¹) none) (some val✝) = map₂ g'... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝¹ : α
val✝ : β
⊢ map₂ f (map₂ g (some val✝¹) (some val✝)) none = map₂ g'... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
case some.some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f✝ : α → β → γ
δ : Type u_1
ε : Type u_2
δ' : Type u_3
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
val✝² : α
val✝¹ : β
val✝ : γ
⊢ map₂ f (map₂ g (some val✝²) (some val✝¹)) (so... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_right_comm] | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b :=
by cases a <;> cases b <;> cases c <;> | Mathlib.Data.Option.NAry.140_0.qAUZsy5CifjOLJ4 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b | Mathlib_Data_Option_NAry |
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
⊢ Option.map g (map₂ f a b) = map₂ f' (Option.map g₁ a) (Option.map g₂ b) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case none
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
⊢ Option.map g (map₂ f none b) = map₂ f' (Option.map g₁ none) (Option.map g₂ b) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case some
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
val✝ : α
⊢ Option.map g (map₂ f (some val✝) b) = map₂ f' (Option.map g₁ (some v... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case none.none
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
⊢ Option.map g (map₂ f none none) = map₂ f' (Option.map g₁ none) (Option.map g₂ none) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_distrib] | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case none.some
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
val✝ : β
⊢ Option.map g (map₂ f none (some val✝)) = map₂ f' (Option.map g₁ none) (Optio... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_distrib] | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case some.none
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
val✝ : α
⊢ Option.map g (map₂ f (some val✝) none) = map₂ f' (Option.map g₁ (some val✝))... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_distrib] | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
case some.some
α : Type u_5
β : Type u_6
γ : Type u_4
f : α → β → γ
c : Option γ
δ : Type u_1
α' : Type u_2
β' : Type u_3
g : γ → δ
f' : α' → β' → δ
g₁ : α → α'
g₂ : β → β'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b)
val✝¹ : α
val✝ : β
⊢ Option.map g (map₂ f (some val✝¹) (some val✝)) = map₂ f' (Option.m... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | simp [h_distrib] | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) :=
by cases a <;> cases b <;> | Mathlib.Data.Option.NAry.146_0.qAUZsy5CifjOLJ4 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) | Mathlib_Data_Option_NAry |
α : Type u_4
β : Type u_5
γ : Type u_3
f : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
g : γ → δ
f' : α' → β → δ
g' : α → α'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g' a) b
⊢ Option.map g (map₂ f a b) = map₂ f' (Option.map g' a) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | /-- Symmetric statement to `Option.map₂_map_left_comm`. -/
theorem map_map₂_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(map₂ f a b).map g = map₂ f' (a.map g') b := by | Mathlib.Data.Option.NAry.157_0.qAUZsy5CifjOLJ4 | /-- Symmetric statement to `Option.map₂_map_left_comm`. -/
theorem map_map₂_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(map₂ f a b).map g = map₂ f' (a.map g') b | Mathlib_Data_Option_NAry |
case none
α : Type u_4
β : Type u_5
γ : Type u_3
f : α → β → γ
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
g : γ → δ
f' : α' → β → δ
g' : α → α'
h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g' a) b
⊢ Option.map g (map₂ f none b) = map₂ f' (Option.map g' none) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases b | /-- Symmetric statement to `Option.map₂_map_left_comm`. -/
theorem map_map₂_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(map₂ f a b).map g = map₂ f' (a.map g') b := by cases a <;> | Mathlib.Data.Option.NAry.157_0.qAUZsy5CifjOLJ4 | /-- Symmetric statement to `Option.map₂_map_left_comm`. -/
theorem map_map₂_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(map₂ f a b).map g = map₂ f' (a.map g') b | Mathlib_Data_Option_NAry |
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