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 succ.prec.intro.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.prec.intro.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases evaln k' _ _ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.prec.intro.succ.none
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.prec.intro.succ.some
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [hg (Nat.pair_lt_pair_right _ lg)] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(List.ma... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have lf := encode_lt_rfind' cf | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(List.ma... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [hg (Nat.pair_lt_pair_right _ lf)] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(List.ma... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases' evaln k cf n with x | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.none
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(Li... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.some
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(Li... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp only [decode_eq_ofNat, Option.some.injEq, Option.some_bind] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.some
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
(Li... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases x | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.some.zero
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [Nat.succ_ne_zero] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.some.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [Nat.succ_ne_zero] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
case succ.rfind'.some.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
∀ {k' : ℕ} {c' : Code} {n : ℕ},
Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →
Nat.Partrec.Code.lup
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
this :
Primrec₂ fun x n =>
let a := ofNat (ℕ × Code) n;
List.map (evaln a.1 a.2) (List.range a.1)
x✝ : (ℕ × Code) × ℕ
k : ℕ
c : Code
n : ℕ
⊢ (Option.bind
(List.get?
(let a := ofNat (ℕ × Code) (encode ((k, c), n).1);
List.map (evaln a.1 a.2) (List.range a.1))
((k, c), n).2)
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [evaln_map] | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primr... | Mathlib.Computability.PartrecCode.1088_0.A3c3Aev6SyIRjCJ | /-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 | Mathlib_Computability_PartrecCode |
c : Code
n x : ℕ
⊢ x ∈ eval c n ↔ x ∈ rfindOpt fun k => evaln k c n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine' evaln_complete.trans (Nat.rfindOpt_mono _).symm | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n :=
Part.ext fun x => by
| Mathlib.Computability.PartrecCode.1157_0.A3c3Aev6SyIRjCJ | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n | Mathlib_Computability_PartrecCode |
c : Code
n x : ℕ
⊢ ∀ {a m n_1 : ℕ}, m ≤ n_1 → a ∈ evaln m c n → a ∈ evaln n_1 c n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | intro a m n hl | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n :=
Part.ext fun x => by
refine' evaln_complete.trans (Nat.rfindOpt_mono _).symm
| Mathlib.Computability.PartrecCode.1157_0.A3c3Aev6SyIRjCJ | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n | Mathlib_Computability_PartrecCode |
c : Code
n✝ x a m n : ℕ
hl : m ≤ n
⊢ a ∈ evaln m c n✝ → a ∈ evaln n c n✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | apply evaln_mono hl | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n :=
Part.ext fun x => by
refine' evaln_complete.trans (Nat.rfindOpt_mono _).symm
intro a m n hl; | Mathlib.Computability.PartrecCode.1157_0.A3c3Aev6SyIRjCJ | theorem eval_eq_rfindOpt (c n) : eval c n = Nat.rfindOpt fun k => evaln k c n | Mathlib_Computability_PartrecCode |
a : Code × ℕ
⊢ (rfindOpt fun b =>
evaln (((a, b).2, (a, b).1.1), (a, b).1.2).1.1 (((a, b).2, (a, b).1.1), (a, b).1.2).1.2
(((a, b).2, (a, b).1.1), (a, b).1.2).2) =
eval a.1 a.2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [eval_eq_rfindOpt] | theorem eval_part : Partrec₂ eval :=
(Partrec.rfindOpt
(evaln_prim.to_comp.comp ((Computable.snd.pair (fst.comp fst)).pair (snd.comp fst))).to₂).of_eq
fun a => by | Mathlib.Computability.PartrecCode.1163_0.A3c3Aev6SyIRjCJ | theorem eval_part : Partrec₂ eval | Mathlib_Computability_PartrecCode |
f : Code → Code
hf : Computable f
g : ℕ → ℕ → Part ℕ :=
fun x y => do
let b ← eval (ofNat Code x) x
eval (ofNat Code b) y
this : Partrec₂ g
cg : Code
eg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.1 p.2) a)
⊢ ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g a n) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [eg] | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c :=
let ... | Mathlib.Computability.PartrecCode.1169_0.A3c3Aev6SyIRjCJ | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c | Mathlib_Computability_PartrecCode |
f : Code → Code
hf : Computable f
g : ℕ → ℕ → Part ℕ :=
fun x y => do
let b ← eval (ofNat Code x) x
eval (ofNat Code b) y
this✝ : Partrec₂ g
cg : Code
eg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.1 p.2) a)
eg' : ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [eF] | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c :=
let ... | Mathlib.Computability.PartrecCode.1169_0.A3c3Aev6SyIRjCJ | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c | Mathlib_Computability_PartrecCode |
f : Code → Code
hf : Computable f
g : ℕ → ℕ → Part ℕ :=
fun x y => do
let b ← eval (ofNat Code x) x
eval (ofNat Code b) y
this✝ : Partrec₂ g
cg : Code
eg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.1 p.2) a)
eg' : ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [eg', eF', Part.map_id'] | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c :=
let ... | Mathlib.Computability.PartrecCode.1169_0.A3c3Aev6SyIRjCJ | /-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c | Mathlib_Computability_PartrecCode |
f : Code → ℕ →. ℕ
hf : Partrec₂ f
cf : Code
ef : eval cf = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => f p.1 p.2) a)
c : Code
e : eval (curry cf (encode c)) = eval c
n : ℕ
⊢ eval c n = f c n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [e.symm, ef, Part.map_id'] | theorem fixed_point₂ {f : Code → ℕ →. ℕ} (hf : Partrec₂ f) : ∃ c : Code, eval c = f c :=
let ⟨cf, ef⟩ := exists_code.1 hf
(fixed_point (curry_prim.comp (_root_.Primrec.const cf) Primrec.encode).to_comp).imp fun c e =>
funext fun n => by | Mathlib.Computability.PartrecCode.1191_0.A3c3Aev6SyIRjCJ | theorem fixed_point₂ {f : Code → ℕ →. ℕ} (hf : Partrec₂ f) : ∃ c : Code, eval c = f c | Mathlib_Computability_PartrecCode |
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
f : G →* H
emb : OpenEmbedding ⇑f
U : Set H
hU : U ∈ nhds... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | apply emb.continuous.tendsto | /-- If a topological group embeds into a nonarchimedean group, then it is nonarchimedean. -/
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have h₁ : f ⁻¹' U ∈ nhds (1 : G) := by
| Mathlib.Topology.Algebra.Nonarchimedean.Basic.69_0.BrtsnGem4TIvd8C | /-- If a topological group embeds into a nonarchimedean group, then it is nonarchimedean. -/
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case a
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
f : G →* H
emb : OpenEmbedding ⇑f
U : Set H
hU : U... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | rwa [f.map_one] | /-- If a topological group embeds into a nonarchimedean group, then it is nonarchimedean. -/
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have h₁ : f ⁻¹' U ∈ nhds (1 : G) := by
apply emb.continuous.tendsto... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.69_0.BrtsnGem4TIvd8C | /-- If a topological group embeds into a nonarchimedean group, then it is nonarchimedean. -/
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
hU : U ∈ nhds 1
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | erw [nhds_prod_eq, Filter.mem_prod_iff] at hU | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
hU : ∃ t₁ ∈ nhds 1, ∃ t₂ ∈ nhds 1, t₁ ×ˢ ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | rcases hU with ⟨U₁, hU₁, U₂, hU₂, h⟩ | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
h... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | cases' is_nonarchimedean _ hU₁ with V hV | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : S... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | cases' is_nonarchimedean _ hU₂ with W hW | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | use V | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | use W | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | rw [Set.prod_subset_iff] | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | intro x hX y hY | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | exact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY) | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.81_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. -/
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains th... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : OpenSubgroup G
h : ... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | refine' Set.Subset.trans (Set.prod_mono _ _) ‹_› | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.99_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case refine'_1
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : Open... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | simp | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.99_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case refine'_2
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : Open... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | simp | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.99_0.BrtsnGem4TIvd8C | /-- An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. -/
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the car... | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U : OpenAddSubgroup R
⊢ ∃ V, ↑V * ↑V ⊆ ↑U | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero,
mul_zero] using U.zero_mem)) | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
| Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U : OpenAddSubgroup R
⊢ 0 ∈ (fun p => p.1 * p.2) ⁻¹' ↑U | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | simpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero,
mul_zero] using U.zero_mem | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
⊢ ∃ V, ↑V * ↑V ⊆ ↑U | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | use V | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
⊢ ↑V * ↑V ⊆ ↑U | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | rintro v ⟨a, b, ha, hb, hv⟩ | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | have hy := H (Set.mk_mem_prod ha hb) | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | simp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | rw [SetLike.mem_coe] | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.1 * p.2) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1... | /-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.... | exact hy | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ :=
prod_self_subset
(IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
(by simpa only [Set.mem_preim... | Mathlib.Topology.Algebra.Nonarchimedean.Basic.143_0.BrtsnGem4TIvd8C | /-- An open subgroup of a nonarchimedean ring contains the square of another one. -/
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U | Mathlib_Topology_Algebra_Nonarchimedean_Basic |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
⊢ ∀ (J : Type) [𝒥 : SmallCategory J] [inst : FinCategory J], HasLimitsOfShape J α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | intro J 𝒥₁ 𝒥₂ | instance (priority := 100) hasFiniteLimits_of_semilatticeInf_orderTop [SemilatticeInf α]
[OrderTop α] : HasFiniteLimits α := ⟨by
| Mathlib.CategoryTheory.Limits.Lattice.54_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J✝ : Type w
inst✝³ : SmallCategory J✝
inst✝² : FinCategory J✝
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
J : Type
𝒥₁ : SmallCategory J
𝒥₂ : FinCategory J
⊢ HasLimitsOfShape J α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact { has_limit := fun F => HasLimit.mk (finiteLimitCone F) } | instance (priority := 100) hasFiniteLimits_of_semilatticeInf_orderTop [SemilatticeInf α]
[OrderTop α] : HasFiniteLimits α := ⟨by
intro J 𝒥₁ 𝒥₂
| Mathlib.CategoryTheory.Limits.Lattice.54_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
⊢ ∀ (J : Type) [𝒥 : SmallCategory J] [inst : FinCategory J], HasColimitsOfShape J α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | intro J 𝒥₁ 𝒥₂ | instance (priority := 100) hasFiniteColimits_of_semilatticeSup_orderBot [SemilatticeSup α]
[OrderBot α] : HasFiniteColimits α := ⟨by
| Mathlib.CategoryTheory.Limits.Lattice.61_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J✝ : Type w
inst✝³ : SmallCategory J✝
inst✝² : FinCategory J✝
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
J : Type
𝒥₁ : SmallCategory J
𝒥₂ : FinCategory J
⊢ HasColimitsOfShape J α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact { has_colimit := fun F => HasColimit.mk (finiteColimitCocone F) } | instance (priority := 100) hasFiniteColimits_of_semilatticeSup_orderBot [SemilatticeSup α]
[OrderBot α] : HasFiniteColimits α := ⟨by
intro J 𝒥₁ 𝒥₂
| Mathlib.CategoryTheory.Limits.Lattice.61_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ ∏ f = Finset.inf Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | trans | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f := by
| Mathlib.CategoryTheory.Limits.Lattice.83_0.76fWBXDnykYGQ2r | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ ∏ f = ?m.33948
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f :... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
(finiteLimitCone (Discrete.functor f)).isLimit).to_eq | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f := by
trans
| Mathlib.CategoryTheory.Limits.Lattice.83_0.76fWBXDnykYGQ2r | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ (finiteLimitCone (Discrete.functor f)).cone.pt = Finset.inf Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | change Finset.univ.inf (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.inf f | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f := by
trans
exact
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
... | Mathlib.CategoryTheory.Limits.Lattice.83_0.76fWBXDnykYGQ2r | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ Finset.inf Finset.univ (f ∘ ⇑(Equiv.toEmbedding discreteEquiv)) = Finset.inf Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | simp only [← Finset.inf_map, Finset.univ_map_equiv_to_embedding] | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f := by
trans
exact
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
... | Mathlib.CategoryTheory.Limits.Lattice.83_0.76fWBXDnykYGQ2r | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ Finset.inf Finset.univ f = Finset.inf Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rfl | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f := by
trans
exact
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
... | Mathlib.CategoryTheory.Limits.Lattice.83_0.76fWBXDnykYGQ2r | /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ f = Fintype.elems.inf f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ ∐ f = Finset.sup Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | trans | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by
| Mathlib.CategoryTheory.Limits.Lattice.97_0.76fWBXDnykYGQ2r | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ ∐ f = ?m.46970
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f :... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact
(IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
(finiteColimitCocone (Discrete.functor f)).isColimit).to_eq | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by
trans
| Mathlib.CategoryTheory.Limits.Lattice.97_0.76fWBXDnykYGQ2r | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ (finiteColimitCocone (Discrete.functor f)).cocone.pt = Finset.sup Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | change Finset.univ.sup (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.sup f | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by
trans
exact
(IsColimit.coconePointUniqueUpToIso (colimit.... | Mathlib.CategoryTheory.Limits.Lattice.97_0.76fWBXDnykYGQ2r | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ Finset.sup Finset.univ (f ∘ ⇑(Equiv.toEmbedding discreteEquiv)) = Finset.sup Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | simp only [← Finset.sup_map, Finset.univ_map_equiv_to_embedding] | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by
trans
exact
(IsColimit.coconePointUniqueUpToIso (colimit.... | Mathlib.CategoryTheory.Limits.Lattice.97_0.76fWBXDnykYGQ2r | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝⁴ : SmallCategory J
inst✝³ : FinCategory J
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
ι : Type u
inst✝ : Fintype ι
f : ι → α
⊢ Finset.sup Finset.univ f = Finset.sup Fintype.elems f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rfl | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by
trans
exact
(IsColimit.coconePointUniqueUpToIso (colimit.... | Mathlib.CategoryTheory.Limits.Lattice.97_0.76fWBXDnykYGQ2r | /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
⊢ HasBinaryProducts α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance | instance (priority := 100) [SemilatticeInf α] [OrderTop α] : HasBinaryProducts α := by
| Mathlib.CategoryTheory.Limits.Lattice.112_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
⊢ ∀ (x y : α), HasLimit (pair x y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α | instance (priority := 100) [SemilatticeInf α] [OrderTop α] : HasBinaryProducts α := by
have : ∀ x y : α, HasLimit (pair x y) := by
| Mathlib.CategoryTheory.Limits.Lattice.112_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
this : (∀ (J : Type u) {𝒥 : SmallCategory J}, FinCategory J → HasLimitsOfShape J α) → HasFiniteLimits α :=
hasFiniteLimits_of_hasFiniteLimits_of_size α
⊢ ∀ (x y : α), HasLimit (pair x y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | infer_instance | instance (priority := 100) [SemilatticeInf α] [OrderTop α] : HasBinaryProducts α := by
have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
| Mathlib.CategoryTheory.Limits.Lattice.112_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
this : ∀ (x y : α), HasLimit (pair x y)
⊢ HasBinaryProducts α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | apply hasBinaryProducts_of_hasLimit_pair | instance (priority := 100) [SemilatticeInf α] [OrderTop α] : HasBinaryProducts α := by
have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance
| Mathlib.CategoryTheory.Limits.Lattice.112_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
x y : α
⊢ limit (pair x y) = Finset.inf Finset.univ (pair x y).toPrefunctor.obj | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [finite_limit_eq_finset_univ_inf (pair.{u} x y)] | /-- The binary product in the category of a `SemilatticeInf` with `OrderTop` is the same as the
infimum.
-/
@[simp]
theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y :=
calc
Limits.prod x y = limit (pair x y) := rfl
_ = Finset.univ.inf (pair x y).obj := by | Mathlib.CategoryTheory.Limits.Lattice.118_0.76fWBXDnykYGQ2r | /-- The binary product in the category of a `SemilatticeInf` with `OrderTop` is the same as the
infimum.
-/
@[simp]
theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
x y : α
⊢ x ⊓ (y ⊓ ⊤) = x ⊓ y | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [inf_top_eq] | /-- The binary product in the category of a `SemilatticeInf` with `OrderTop` is the same as the
infimum.
-/
@[simp]
theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y :=
calc
Limits.prod x y = limit (pair x y) := rfl
_ = Finset.univ.inf (pair x y).obj := by rw [finite_limi... | Mathlib.CategoryTheory.Limits.Lattice.118_0.76fWBXDnykYGQ2r | /-- The binary product in the category of a `SemilatticeInf` with `OrderTop` is the same as the
infimum.
-/
@[simp]
theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
⊢ HasBinaryCoproducts α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | have : ∀ x y : α, HasColimit (pair x y) := by
letI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} α
infer_instance | instance (priority := 100) [SemilatticeSup α] [OrderBot α] : HasBinaryCoproducts α := by
| Mathlib.CategoryTheory.Limits.Lattice.132_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
⊢ ∀ (x y : α), HasColimit (pair x y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | letI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} α | instance (priority := 100) [SemilatticeSup α] [OrderBot α] : HasBinaryCoproducts α := by
have : ∀ x y : α, HasColimit (pair x y) := by
| Mathlib.CategoryTheory.Limits.Lattice.132_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
this : (∀ (J : Type u) {𝒥 : SmallCategory J}, FinCategory J → HasColimitsOfShape J α) → HasFiniteColimits α :=
hasFiniteColimits_of_hasFiniteColimits_of_size α
⊢ ∀ (x y : α), HasColimit (pair x y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | infer_instance | instance (priority := 100) [SemilatticeSup α] [OrderBot α] : HasBinaryCoproducts α := by
have : ∀ x y : α, HasColimit (pair x y) := by
letI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} α
| Mathlib.CategoryTheory.Limits.Lattice.132_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
this : ∀ (x y : α), HasColimit (pair x y)
⊢ HasBinaryCoproducts α | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | apply hasBinaryCoproducts_of_hasColimit_pair | instance (priority := 100) [SemilatticeSup α] [OrderBot α] : HasBinaryCoproducts α := by
have : ∀ x y : α, HasColimit (pair x y) := by
letI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} α
infer_instance
| Mathlib.CategoryTheory.Limits.Lattice.132_0.76fWBXDnykYGQ2r | instance (priority | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y : α
⊢ colimit (pair x y) = Finset.sup Finset.univ (pair x y).toPrefunctor.obj | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [finite_colimit_eq_finset_univ_sup (pair x y)] | /-- The binary coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
@[simp]
theorem coprod_eq_sup [SemilatticeSup α] [OrderBot α] (x y : α) : Limits.coprod x y = x ⊔ y :=
calc
Limits.coprod x y = colimit (pair x y) := rfl
_ = Finset.univ.sup (pair x y).obj := by | Mathlib.CategoryTheory.Limits.Lattice.138_0.76fWBXDnykYGQ2r | /-- The binary coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
@[simp]
theorem coprod_eq_sup [SemilatticeSup α] [OrderBot α] (x y : α) : Limits.coprod x y = x ⊔ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y : α
⊢ x ⊔ (y ⊔ ⊥) = x ⊔ y | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [sup_bot_eq] | /-- The binary coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
@[simp]
theorem coprod_eq_sup [SemilatticeSup α] [OrderBot α] (x y : α) : Limits.coprod x y = x ⊔ y :=
calc
Limits.coprod x y = colimit (pair x y) := rfl
_ = Finset.univ.sup (pair x y).obj := by rw [... | Mathlib.CategoryTheory.Limits.Lattice.138_0.76fWBXDnykYGQ2r | /-- The binary coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
@[simp]
theorem coprod_eq_sup [SemilatticeSup α] [OrderBot α] (x y : α) : Limits.coprod x y = x ⊔ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
x y z : α
f : x ⟶ z
g : y ⟶ z
⊢ limit (cospan f g) = Finset.inf Finset.univ (cospan f g).toPrefunctor.obj | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [finite_limit_eq_finset_univ_inf] | /-- The pullback in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum
over the objects.
-/
@[simp]
theorem pullback_eq_inf [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
pullback f g = x ⊓ y :=
calc
pullback f g = limit (cospan f g) := rfl
_ = Finset.univ... | Mathlib.CategoryTheory.Limits.Lattice.151_0.76fWBXDnykYGQ2r | /-- The pullback in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum
over the objects.
-/
@[simp]
theorem pullback_eq_inf [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
pullback f g = x ⊓ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
x y z : α
f : x ⟶ z
g : y ⟶ z
⊢ z ⊓ (x ⊓ (y ⊓ ⊤)) = z ⊓ (x ⊓ y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [inf_top_eq] | /-- The pullback in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum
over the objects.
-/
@[simp]
theorem pullback_eq_inf [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
pullback f g = x ⊓ y :=
calc
pullback f g = limit (cospan f g) := rfl
_ = Finset.univ... | Mathlib.CategoryTheory.Limits.Lattice.151_0.76fWBXDnykYGQ2r | /-- The pullback in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum
over the objects.
-/
@[simp]
theorem pullback_eq_inf [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
pullback f g = x ⊓ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y
⊢ colimit (span f g) = Finset.sup Finset.univ (span f g).toPrefunctor.obj | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [finite_colimit_eq_finset_univ_sup] | /-- The pushout in the category of a `SemilatticeSup` with `OrderBot` is the same as the supremum
over the objects.
-/
@[simp]
theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y :=
calc
pushout f g = colimit (span f g) := rfl
_ = Finset.univ.su... | Mathlib.CategoryTheory.Limits.Lattice.165_0.76fWBXDnykYGQ2r | /-- The pushout in the category of a `SemilatticeSup` with `OrderBot` is the same as the supremum
over the objects.
-/
@[simp]
theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y
⊢ z ⊔ (x ⊔ (y ⊔ ⊥)) = z ⊔ (x ⊔ y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rw [sup_bot_eq] | /-- The pushout in the category of a `SemilatticeSup` with `OrderBot` is the same as the supremum
over the objects.
-/
@[simp]
theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y :=
calc
pushout f g = colimit (span f g) := rfl
_ = Finset.univ.su... | Mathlib.CategoryTheory.Limits.Lattice.165_0.76fWBXDnykYGQ2r | /-- The pushout in the category of a `SemilatticeSup` with `OrderBot` is the same as the supremum
over the objects.
-/
@[simp]
theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
inst✝¹ : CompleteLattice α
J : Type u
inst✝ : SmallCategory J
F : J ⥤ α
s : Cone F
⊢ ∀ b ∈ Set.range F.obj, s.pt ≤ b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rintro _ ⟨j, rfl⟩ | /-- The limit cone over any functor into a complete lattice.
-/
def limitCone (F : J ⥤ α) : LimitCone F where
cone :=
{ pt := iInf F.obj
π := { app := fun j => homOfLE (CompleteLattice.sInf_le _ _ (Set.mem_range_self _)) } }
isLimit :=
{ lift := fun s =>
homOfLE (CompleteLattice.le_sInf _ _ (b... | Mathlib.CategoryTheory.Limits.Lattice.185_0.76fWBXDnykYGQ2r | /-- The limit cone over any functor into a complete lattice.
-/
def limitCone (F : J ⥤ α) : LimitCone F where
cone | Mathlib_CategoryTheory_Limits_Lattice |
case intro
α : Type u
inst✝¹ : CompleteLattice α
J : Type u
inst✝ : SmallCategory J
F : J ⥤ α
s : Cone F
j : J
⊢ s.pt ≤ F.obj j | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact (s.π.app j).le | /-- The limit cone over any functor into a complete lattice.
-/
def limitCone (F : J ⥤ α) : LimitCone F where
cone :=
{ pt := iInf F.obj
π := { app := fun j => homOfLE (CompleteLattice.sInf_le _ _ (Set.mem_range_self _)) } }
isLimit :=
{ lift := fun s =>
homOfLE (CompleteLattice.le_sInf _ _ (b... | Mathlib.CategoryTheory.Limits.Lattice.185_0.76fWBXDnykYGQ2r | /-- The limit cone over any functor into a complete lattice.
-/
def limitCone (F : J ⥤ α) : LimitCone F where
cone | Mathlib_CategoryTheory_Limits_Lattice |
α : Type u
inst✝¹ : CompleteLattice α
J : Type u
inst✝ : SmallCategory J
F : J ⥤ α
s : Cocone F
⊢ ∀ b ∈ Set.range F.obj, b ≤ s.pt | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | rintro _ ⟨j, rfl⟩ | /-- The colimit cocone over any functor into a complete lattice.
-/
def colimitCocone (F : J ⥤ α) : ColimitCocone F where
cocone :=
{ pt := iSup F.obj
ι := { app := fun j => homOfLE (CompleteLattice.le_sSup _ _ (Set.mem_range_self _)) } }
isColimit :=
{ desc := fun s =>
homOfLE (CompleteLattic... | Mathlib.CategoryTheory.Limits.Lattice.196_0.76fWBXDnykYGQ2r | /-- The colimit cocone over any functor into a complete lattice.
-/
def colimitCocone (F : J ⥤ α) : ColimitCocone F where
cocone | Mathlib_CategoryTheory_Limits_Lattice |
case intro
α : Type u
inst✝¹ : CompleteLattice α
J : Type u
inst✝ : SmallCategory J
F : J ⥤ α
s : Cocone F
j : J
⊢ F.obj j ≤ s.pt | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Fintype.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Categor... | exact (s.ι.app j).le | /-- The colimit cocone over any functor into a complete lattice.
-/
def colimitCocone (F : J ⥤ α) : ColimitCocone F where
cocone :=
{ pt := iSup F.obj
ι := { app := fun j => homOfLE (CompleteLattice.le_sSup _ _ (Set.mem_range_self _)) } }
isColimit :=
{ desc := fun s =>
homOfLE (CompleteLattic... | Mathlib.CategoryTheory.Limits.Lattice.196_0.76fWBXDnykYGQ2r | /-- The colimit cocone over any functor into a complete lattice.
-/
def colimitCocone (F : J ⥤ α) : ColimitCocone F where
cocone | Mathlib_CategoryTheory_Limits_Lattice |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | cases f | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f := f.toLinearEquiv
inv f := f.toLinearEquiv.symm
left_inv f := f.toLinearEquiv.left_inv
right_inv f := f.toLinearEquiv.right_inv
coe_injective' f g := by | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.55_0.YOl7VzX5AOw0Rpu | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case mk
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | cases g | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f := f.toLinearEquiv
inv f := f.toLinearEquiv.symm
left_inv f := f.toLinearEquiv.left_inv
right_inv f := f.toLinearEquiv.right_inv
coe_injective' f g := by cases f; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.55_0.YOl7VzX5AOw0Rpu | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case mk.mk
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Modul... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | simp (config := {contextual := true}) | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f := f.toLinearEquiv
inv f := f.toLinearEquiv.symm
left_inv f := f.toLinearEquiv.left_inv
right_inv f := f.toLinearEquiv.right_inv
coe_injective' f g := by cases f; cases g; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.55_0.YOl7VzX5AOw0Rpu | instance : LinearEquivClass (Q₁.IsometryEquiv Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | intro m | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ :=
{ (f : M₁ ≃ₗ[R] M₂).symm with
map_app' := by | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.87_0.YOl7VzX5AOw0Rpu | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | rw [← f.map_app] | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ :=
{ (f : M₁ ≃ₗ[R] M₂).symm with
map_app' := by intro m; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.87_0.YOl7VzX5AOw0Rpu | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | congr | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ :=
{ (f : M₁ ≃ₗ[R] M₂).symm with
map_app' := by intro m; rw [← f.map_app]; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.87_0.YOl7VzX5AOw0Rpu | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case h.e_6.h
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Mod... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | exact f.toLinearEquiv.apply_symm_apply m | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ :=
{ (f : M₁ ≃ₗ[R] M₂).symm with
map_app' := by intro m; rw [← f.map_app]; congr; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.87_0.YOl7VzX5AOw0Rpu | /-- The inverse isometric equivalence of an isometric equivalence between two quadratic forms. -/
@[symm]
def symm (f : Q₁.IsometryEquiv Q₂) : Q₂.IsometryEquiv Q₁ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | intro m | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ :=
{ (f : M₁ ≃ₗ[R] M₂).trans (g : M₂ ≃ₗ[R] M₃) with
map_app' := by | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.94_0.YOl7VzX5AOw0Rpu | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | rw [← f.map_app, ← g.map_app] | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ :=
{ (f : M₁ ≃ₗ[R] M₂).trans (g : M₂ ≃ₗ[R] M₃) with
map_app' := by intro m; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.94_0.YOl7VzX5AOw0Rpu | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : AddCommMonoid M₃
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | rfl | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ :=
{ (f : M₁ ≃ₗ[R] M₂).trans (g : M₂ ≃ₗ[R] M₃) with
map_app' := by intro m; rw [← f.map_app, ← g.map_app]; | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.94_0.YOl7VzX5AOw0Rpu | /-- The composition of two isometric equivalences between quadratic forms. -/
@[trans]
def trans (f : Q₁.IsometryEquiv Q₂) (g : Q₂.IsometryEquiv Q₃) : Q₁.IsometryEquiv Q₃ | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M₂
inst✝¹ : Module R M₃
ins... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | intro | /-- A quadratic form composed with a `LinearEquiv` is isometric to itself. -/
def isometryEquivOfCompLinearEquiv (Q : QuadraticForm R M) (f : M₁ ≃ₗ[R] M) :
Q.IsometryEquiv (Q.comp (f : M₁ →ₗ[R] M)) :=
{ f.symm with
map_app' := by
| Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.132_0.YOl7VzX5AOw0Rpu | /-- A quadratic form composed with a `LinearEquiv` is isometric to itself. -/
def isometryEquivOfCompLinearEquiv (Q : QuadraticForm R M) (f : M₁ ≃ₗ[R] M) :
Q.IsometryEquiv (Q.comp (f : M₁ →ₗ[R] M)) | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M₂
inst✝¹ : Module R M₃
ins... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | simp only [comp_apply, LinearEquiv.coe_coe, LinearEquiv.toFun_eq_coe,
LinearEquiv.apply_symm_apply, f.apply_symm_apply] | /-- A quadratic form composed with a `LinearEquiv` is isometric to itself. -/
def isometryEquivOfCompLinearEquiv (Q : QuadraticForm R M) (f : M₁ ≃ₗ[R] M) :
Q.IsometryEquiv (Q.comp (f : M₁ →ₗ[R] M)) :=
{ f.symm with
map_app' := by
intro
| Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.132_0.YOl7VzX5AOw0Rpu | /-- A quadratic form composed with a `LinearEquiv` is isometric to itself. -/
def isometryEquivOfCompLinearEquiv (Q : QuadraticForm R M) (f : M₁ ≃ₗ[R] M) :
Q.IsometryEquiv (Q.comp (f : M₁ →ₗ[R] M)) | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : Module R M
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
inst✝⁵ : Module R M₃... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | let iso := Q.isometryEquivBasisRepr v | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R := K) Q).iIsOrtho v) :
Q.IsometryEquiv (weighte... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.150_0.YOl7VzX5AOw0Rpu | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : Module R M
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
inst✝⁵ : Module R M₃... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | refine' ⟨iso, fun m => _⟩ | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R := K) Q).iIsOrtho v) :
Q.IsometryEquiv (weighte... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.150_0.YOl7VzX5AOw0Rpu | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : Module R M
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
inst✝⁵ : Module R M₃... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | convert iso.map_app m | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R := K) Q).iIsOrtho v) :
Q.IsometryEquiv (weighte... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.150_0.YOl7VzX5AOw0Rpu | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case h.e'_2.h.e'_5
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : Module R M
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
i... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | rw [basisRepr_eq_of_iIsOrtho _ _ hv₁] | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R := K) Q).iIsOrtho v) :
Q.IsometryEquiv (weighte... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.150_0.YOl7VzX5AOw0Rpu | /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
squares. -/
noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
(v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
(hv₁ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid M₁
inst✝¹¹ : AddCommMonoid M₂
inst✝¹⁰ : AddCommMonoid M₃
inst✝⁹ : Module R M
inst✝⁸ : Module R M₁
inst✝⁷ : Module R M₂
inst✝⁶ : Module R M... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | obtain ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm K Q) | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R := K) Q).Nondegenerate) :
∃ w : Fin (FiniteDimensional.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
| Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.172_0.YOl7VzX5AOw0Rpu | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case intro
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid M₁
inst✝¹¹ : AddCommMonoid M₂
inst✝¹⁰ : AddCommMonoid M₃
inst✝⁹ : Module R M
inst✝⁸ : Module R M₁
inst✝⁷ : Module R M₂
inst✝⁶ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | have hv₂ := hv₁.not_isOrtho_basis_self_of_nondegenerate hQ | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R := K) Q).Nondegenerate) :
∃ w : Fin (FiniteDimensional.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
obtain ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm K Q)
| Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.172_0.YOl7VzX5AOw0Rpu | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case intro
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid M₁
inst✝¹¹ : AddCommMonoid M₂
inst✝¹⁰ : AddCommMonoid M₃
inst✝⁹ : Module R M
inst✝⁸ : Module R M₁
inst✝⁷ : Module R M₂
inst✝⁶ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | simp_rw [IsOrtho, associated_eq_self_apply] at hv₂ | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R := K) Q).Nondegenerate) :
∃ w : Fin (FiniteDimensional.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
obtain ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm K Q)
have hv₂ := hv₁.... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.172_0.YOl7VzX5AOw0Rpu | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
case intro
ι : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
V : Type u_8
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid M₁
inst✝¹¹ : AddCommMonoid M₂
inst✝¹⁰ : AddCommMonoid M₃
inst✝⁹ : Module R M
inst✝⁸ : Module R M₁
inst✝⁷ : Module R M₂
inst✝⁶ :... | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
#align_import linear_algebra.quadratic_form.isometry from... | exact ⟨fun i => Units.mk0 _ (hv₂ i), ⟨Q.isometryEquivWeightedSumSquares v hv₁⟩⟩ | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R := K) Q).Nondegenerate) :
∃ w : Fin (FiniteDimensional.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
obtain ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm K Q)
have hv₂ := hv₁.... | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv.172_0.YOl7VzX5AOw0Rpu | theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
(hQ : (associated (R | Mathlib_LinearAlgebra_QuadraticForm_IsometryEquiv |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ 𝟙 X✝ ≫ f = f | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp (config := { unfoldPartialApp := true }) [dite_comp] | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by | Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ f ≫ 𝟙 Y✝ = f | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp (config := { unfoldPartialApp := true }) [comp_dite] | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by simp (config := { unfoldPartialApp := true }) [dite_comp]
comp_id f := by | Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
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