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 insert
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
⊢ (r c b ∨ r... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | apply or_congr Iff.rfl | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
⊢ (r c (f a)... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | constructor | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert.mp
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
⊢ (r c (f... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | rintro (h₁ | ⟨x, hx, h₂⟩) | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert.mp.inl
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
h₁ : ... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | use a | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
h₁ : r c (f a)
⊢ ... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | simp [h₁] | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert.mp.inr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | refine' ⟨x, by simp [hx], h₂⟩ | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
x : α
hx : x ∈ s
h₂ : r ... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | simp [hx] | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
⊢ (∃ x ∈... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | rintro ⟨x, hx, h⟩ | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
case insert.mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | exact (mem_insert.mp hx).imp (fun hx => by rwa [hx] at h) (fun hx => ⟨x, hx, h⟩) | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha✝ : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
r : β → β → Prop
hr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z
c : β
a : α
s : Finset α
ha : a ∉ s
IH : r c (fold op b f s) ↔ r c b ∨ ∃ x ∈ s, r c (f x)
x : α
hx✝ : x ∈ insert a... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | rwa [hx] at h | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) := by
classical
induction' s using Finset.induction_on with a s ha IH
· simp
rw [Finset.fold_insert ha, hr, IH, ← or_assoc, @or_comm (r c (f a)), or_a... | Mathlib.Data.Finset.Fold.182_0.vMyQI0nR4eRmtxs | theorem fold_op_rel_iff_or {r : β → β → Prop} (hr : ∀ {x y z}, r x (op y z) ↔ r x y ∨ r x z)
{c : β} : r c (s.fold op b f) ↔ r c b ∨ ∃ x ∈ s, r c (f x) | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a : α
inst✝ : DecidableEq α
s : Finset α
⊢ fold (fun x x_1 => x ∪ x_1) ∅ singleton s = s | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | induction' s using Finset.induction_on with a s has ih | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s := by
| Mathlib.Data.Finset.Fold.198_0.vMyQI0nR4eRmtxs | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s | Mathlib_Data_Finset_Fold |
case empty
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : DecidableEq α
⊢ fold (fun x x_1 => x ∪ x_1) ∅ singleton ∅ = ∅ | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | simp only [fold_empty] | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s := by
induction' s using Finset.induction_on with a s has ih
· | Mathlib.Data.Finset.Fold.198_0.vMyQI0nR4eRmtxs | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s | Mathlib_Data_Finset_Fold |
case insert
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝ : α
inst✝ : DecidableEq α
a : α
s : Finset α
has : a ∉ s
ih : fold (fun x x_1 => x ∪ x_1) ∅ singleton s = s
⊢ fold (fun x x_1 => x ∪ x_1) ∅ singleton (insert a s) = insert a... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | rw [fold_insert has, ih, insert_eq] | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s := by
induction' s using Finset.induction_on with a s has ih
· simp only [fold_empty]
· | Mathlib.Data.Finset.Fold.198_0.vMyQI0nR4eRmtxs | @[simp]
theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ fold min b f s ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ ≥ _ ↔ _ | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c := by
| Mathlib.Data.Finset.Fold.219_0.vMyQI0nR4eRmtxs | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ c ≥ fold min b f s ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | apply fold_op_rel_iff_or | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
| Mathlib.Data.Finset.Fold.219_0.vMyQI0nR4eRmtxs | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ ∀ {x y z : β}, x ≥ min y z ↔ x ≥ y ∨ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | intro x y z | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_or
| Mathlib.Data.Finset.Fold.219_0.vMyQI0nR4eRmtxs | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ x ≥ min y z ↔ x ≥ y ∨ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ ≤ _ ↔ _ | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_or
intro x y z
| Mathlib.Data.Finset.Fold.219_0.vMyQI0nR4eRmtxs | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ min y z ≤ x ↔ x ≥ y ∨ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | exact min_le_iff | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_or
intro x y z
show _ ≤ _ ↔ _
| Mathlib.Data.Finset.Fold.219_0.vMyQI0nR4eRmtxs | theorem fold_min_le : s.fold min b f ≤ c ↔ b ≤ c ∨ ∃ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ fold min b f s < c ↔ b < c ∨ ∃ x ∈ s, f x < c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ > _ ↔ _ | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c := by
| Mathlib.Data.Finset.Fold.231_0.vMyQI0nR4eRmtxs | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ c > fold min b f s ↔ b < c ∨ ∃ x ∈ s, f x < c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | apply fold_op_rel_iff_or | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c := by
show _ > _ ↔ _
| Mathlib.Data.Finset.Fold.231_0.vMyQI0nR4eRmtxs | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ ∀ {x y z : β}, x > min y z ↔ x > y ∨ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | intro x y z | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_or
| Mathlib.Data.Finset.Fold.231_0.vMyQI0nR4eRmtxs | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ x > min y z ↔ x > y ∨ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ < _ ↔ _ | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_or
intro x y z
| Mathlib.Data.Finset.Fold.231_0.vMyQI0nR4eRmtxs | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ min y z < x ↔ x > y ∨ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | exact min_lt_iff | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_or
intro x y z
show _ < _ ↔ _
| Mathlib.Data.Finset.Fold.231_0.vMyQI0nR4eRmtxs | theorem fold_min_lt : s.fold min b f < c ↔ b < c ∨ ∃ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ fold max b f s ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ ≥ _ ↔ _ | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c := by
| Mathlib.Data.Finset.Fold.239_0.vMyQI0nR4eRmtxs | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ c ≥ fold max b f s ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | apply fold_op_rel_iff_and | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
| Mathlib.Data.Finset.Fold.239_0.vMyQI0nR4eRmtxs | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ ∀ {x y z : β}, x ≥ max y z ↔ x ≥ y ∧ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | intro x y z | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_and
| Mathlib.Data.Finset.Fold.239_0.vMyQI0nR4eRmtxs | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ x ≥ max y z ↔ x ≥ y ∧ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ ≤ _ ↔ _ | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_and
intro x y z
| Mathlib.Data.Finset.Fold.239_0.vMyQI0nR4eRmtxs | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ max y z ≤ x ↔ x ≥ y ∧ x ≥ z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | exact max_le_iff | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c := by
show _ ≥ _ ↔ _
apply fold_op_rel_iff_and
intro x y z
show _ ≤ _ ↔ _
| Mathlib.Data.Finset.Fold.239_0.vMyQI0nR4eRmtxs | theorem fold_max_le : s.fold max b f ≤ c ↔ b ≤ c ∧ ∀ x ∈ s, f x ≤ c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ fold max b f s < c ↔ b < c ∧ ∀ x ∈ s, f x < c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ > _ ↔ _ | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c := by
| Mathlib.Data.Finset.Fold.251_0.vMyQI0nR4eRmtxs | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ c > fold max b f s ↔ b < c ∧ ∀ x ∈ s, f x < c | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | apply fold_op_rel_iff_and | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c := by
show _ > _ ↔ _
| Mathlib.Data.Finset.Fold.251_0.vMyQI0nR4eRmtxs | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c : β
⊢ ∀ {x y z : β}, x > max y z ↔ x > y ∧ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | intro x y z | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_and
| Mathlib.Data.Finset.Fold.251_0.vMyQI0nR4eRmtxs | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ x > max y z ↔ x > y ∧ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | show _ < _ ↔ _ | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_and
intro x y z
| Mathlib.Data.Finset.Fold.251_0.vMyQI0nR4eRmtxs | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
case hr
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝ : LinearOrder β
c x y z : β
⊢ max y z < x ↔ x > y ∧ x > z | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | exact max_lt_iff | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c := by
show _ > _ ↔ _
apply fold_op_rel_iff_and
intro x y z
show _ < _ ↔ _
| Mathlib.Data.Finset.Fold.251_0.vMyQI0nR4eRmtxs | theorem fold_max_lt : s.fold max b f < c ↔ b < c ∧ ∀ x ∈ s, f x < c | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a : α
inst✝² : LinearOrder β
c : β
inst✝¹ : Add β
inst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n : WithBot β
s : Finset α
⊢ fold max ⊥ (fun x => ↑(f... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | classical
induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right] | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by
| Mathlib.Data.Finset.Fold.263_0.vMyQI0nR4eRmtxs | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n | Mathlib_Data_Finset_Fold |
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a : α
inst✝² : LinearOrder β
c : β
inst✝¹ : Add β
inst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n : WithBot β
s : Finset α
⊢ fold max ⊥ (fun x => ↑(f... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | induction' s using Finset.induction_on with a s _ ih | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by
classical
| Mathlib.Data.Finset.Fold.263_0.vMyQI0nR4eRmtxs | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n | Mathlib_Data_Finset_Fold |
case empty
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s : Finset α
a : α
inst✝² : LinearOrder β
c : β
inst✝¹ : Add β
inst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n : WithBot β
⊢ fold max ⊥ (fun x => ↑(f x)... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | simp [*, max_add_add_right] | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by
classical
induction' s using Finset.induction_on with a s _ ih <;> | Mathlib.Data.Finset.Fold.263_0.vMyQI0nR4eRmtxs | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n | Mathlib_Data_Finset_Fold |
case insert
α : Type u_1
β : Type u_2
γ : Type u_3
op : β → β → β
hc : IsCommutative β op
ha : IsAssociative β op
f : α → β
b : β
s✝ : Finset α
a✝¹ : α
inst✝² : LinearOrder β
c : β
inst✝¹ : Add β
inst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n : WithBot β
a : α
s : Finset α
a✝ : a... | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-commun... | simp [*, max_add_add_right] | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by
classical
induction' s using Finset.induction_on with a s _ ih <;> | Mathlib.Data.Finset.Fold.263_0.vMyQI0nR4eRmtxs | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n | Mathlib_Data_Finset_Fold |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
⊢ (zero ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := hF.1.some | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
| Mathlib.CategoryTheory.Shift.Induced.54_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have h : whiskerLeft F (zero F s i hF).hom = _ :=
((whiskeringLeft C D D).obj F).image_preimage _ | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
letI := hF.1.some
| Mathlib.CategoryTheory.Shift.Induced.54_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | exact (NatTrans.congr_app h X).trans (by simp) | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
letI := hF.1.some
have h : whiskerLeft F (zero F s i hF).hom = _ :=
((whiskeringLeft C D D).obj F).image_preimage _
| Mathlib.CategoryTheory.Shift.Induced.54_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
letI := hF.1.some
have h : whiskerLeft F (zero F s i hF).hom = _ :=
((whiskeringLeft C D D).obj F).image_preimage _
exact (NatTrans.congr_app h X).trans (by | Mathlib.CategoryTheory.Shift.Induced.54_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i hF).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
⊢ (zero ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := hF.1.some | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
| Mathlib.CategoryTheory.Shift.Induced.63_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have h : whiskerLeft F (zero F s i hF).inv = _ :=
((whiskeringLeft C D D).obj F).image_preimage _ | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
letI := hF.1.some
| Mathlib.CategoryTheory.Shift.Induced.63_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | exact (NatTrans.congr_app h X).trans (by simp) | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
letI := hF.1.some
have h : whiskerLeft F (zero F s i hF).inv = _ :=
((whiskeringLeft C D D).obj F).image_preimage _
| Mathlib.CategoryTheory.Shift.Induced.63_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : F... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
letI := hF.1.some
have h : whiskerLeft F (zero F s i hF).inv = _ :=
((whiskeringLeft C D D).obj F).image_preimage _
exact (NatTrans.congr_app h X).trans (by | Mathlib.CategoryTheory.Shift.Induced.63_0.hEyfGnVpBGMswTi | @[simp]
lemma zero_inv_app_obj (X : C) :
(zero F s i hF).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := hF.1.some | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) := by
| Mathlib.CategoryTheory.Shift.Induced.72_0.hEyfGnVpBGMswTi | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have h : whiskerLeft F (add F s i hF a b).hom = _ :=
((whiskeringLeft C D D).obj F).image_preimage _ | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) := by
letI := hF.1.some
| Mathlib.CategoryTheory.Shift.Induced.72_0.hEyfGnVpBGMswTi | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | exact (NatTrans.congr_app h X).trans (by simp) | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) := by
letI := hF.1.some
have h : whiskerLeft F (add F s i hF a b).hom ... | Mathlib.CategoryTheory.Shift.Induced.72_0.hEyfGnVpBGMswTi | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_5
D : Type u_2
inst✝³ : Category.{u_4, u_5} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) := by
letI := hF.1.some
have h : whiskerLeft F (add F s i hF a b).hom ... | Mathlib.CategoryTheory.Shift.Induced.72_0.hEyfGnVpBGMswTi | @[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i hF a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := hF.1.some | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X := by
| Mathlib.CategoryTheory.Shift.Induced.82_0.hEyfGnVpBGMswTi | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have h : whiskerLeft F (add F s i hF a b).inv = _ :=
((whiskeringLeft C D D).obj F).image_preimage _ | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X := by
letI := hF.1.some
| Mathlib.CategoryTheory.Shift.Induced.82_0.hEyfGnVpBGMswTi | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | exact (NatTrans.congr_app h X).trans (by simp) | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X := by
letI := hF.1.some
have h : whiskerLeft F (add F s i hF a b).inv ... | Mathlib.CategoryTheory.Shift.Induced.82_0.hEyfGnVpBGMswTi | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X := by
letI := hF.1.some
have h : whiskerLeft F (add F s i hF a b).inv ... | Mathlib.CategoryTheory.Shift.Induced.82_0.hEyfGnVpBGMswTi | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i hF a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have := hF.2 | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | suffices (Induced.add F s i hF (m₁ + m₂) m₃).hom ≫
whiskerRight (Induced.add F s i hF m₁ m₂).hom (s m₃) =
eqToHom (by rw [add_assoc]) ≫ (Induced.add F s i hF m₁ (m₂ + m₃)).hom ≫
whiskerLeft (s m₁) (Induced.add F s i hF m₂ m₃).hom by
intro X
simpa using NatTrans.... | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rw [add_assoc] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | intro X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simpa using NatTrans.congr_app this X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | apply ((whiskeringLeft C D D).obj F).map_injective | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | ext X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | dsimp | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have eq := F.congr_map (shiftFunctorAdd'_assoc_hom_app
m₁ m₂ m₃ _ _ (m₁+m₂+m₃) rfl rfl rfl X) | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [shiftFunctorAdd'_eq_shiftFunctorAdd] at eq | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [Functor.comp_obj, Functor.map_comp, shiftFunctorAdd',
Iso.trans_hom, eqToIso.hom, NatTrans.comp_app, eqToHom_app,
Category.assoc] at eq | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rw [← cancel_mono ((s m₃).map ((s m₂).map ((i m₁).hom.app X)))] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [Induced.add_hom_app_obj, Category.assoc, Functor.map_comp] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | slice_lhs 4 5 =>
erw [← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [Category.id_comp] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | slice_lhs 6 7 =>
erw [← Functor.map_comp, ← Functor.map_comp, Iso.inv_hom_id_app,
(s m₂).map_id, (s m₃).map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Fait... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, ← Functor.map_comp, Iso.inv_hom_id_app,
(s m₂).map_id, (s m₃).map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Fait... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, ← Functor.map_comp, Iso.inv_hom_id_app,
(s m₂).map_id, (s m₃).map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.a.a.a.a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Fait... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp, ← Functor.map_comp, Iso.inv_hom_id_app,
(s m₂).map_id, (s m₃).map_id] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [Category.comp_id, ← NatTrans.naturality_assoc, reassoc_of% eq,
dcongr_arg (fun a => (i a).hom.app X) (add_assoc m₁ m₂ m₃).symm] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [Functor.comp_obj, eqToHom_map, eqToHom_app, NatTrans.naturality_assoc,
Induced.add_hom_app_obj, Functor.comp_map, Category.assoc, Iso.inv_hom_id_app_assoc,
eqToHom_trans_assoc, eqToHom_refl, Category.id_comp, Category.comp_id,
← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_... | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have := hF.2 | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | suffices (Induced.add F s i hF 0 n).hom =
eqToHom (by rw [zero_add]; rfl) ≫ whiskerRight (Induced.zero F s i hF).inv (s n) by
intro X
simpa using NatTrans.congr_app this X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rw [zero_add] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rfl | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | intro X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simpa using NatTrans.congr_app this X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | apply ((whiskeringLeft C D D).obj F).map_injective | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | ext X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have eq := dcongr_arg (fun a => (i a).hom.app X) (zero_add n) | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | dsimp | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [Induced.add_hom_app_obj, eq, shiftFunctorAdd_zero_add_hom_app,
Functor.map_comp, eqToHom_map, Category.assoc, eqToHom_trans_assoc,
eqToHom_refl, Category.id_comp, eqToHom_app, Induced.zero_inv_app_obj] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← NatTrans.naturality_assoc, Iso.hom_inv_id_app_assoc] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rfl | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | have := hF.2 | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | suffices (Induced.add F s i hF n 0).hom =
eqToHom (by rw [add_zero]; rfl) ≫ whiskerLeft (s n) (Induced.zero F s i hF).inv by
intro X
simpa using NatTrans.congr_app this X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rw [add_zero] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rfl | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | intro X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simpa using NatTrans.congr_app this X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeri... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | apply ((whiskeringLeft C D D).obj F).map_injective | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | ext X | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | dsimp | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [Induced.add_hom_app_obj, dcongr_arg (fun a => (i a).hom.app X) (add_zero n),
← cancel_mono ((s 0).map ((i n).hom.app X)), Category.assoc,
Category.assoc, Category.assoc, Category.assoc, Category.assoc,
Category.assoc, ← (s 0).map_comp, Iso.inv_hom_id_app, Functor.map_id, Category.comp... | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
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