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 |
|---|---|---|---|---|---|---|
⊢ ∀ (x : Γ'),
x ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | intro | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
x✝ : Γ'
⊢ x✝ ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | cases_type* Γ' Bool | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case blank
⊢ blank ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case bit.false
⊢ bit false ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case bit.true
⊢ bit true ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case bra
⊢ bra ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case ket
⊢ ket ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
case comma
⊢ comma ∈
{ val := {blank, bit true, bit false, bra, ket, comma},
nodup := (_ : Multiset.Nodup {blank, bit true, bit false, bra, ket, comma}) } | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | decide | instance Γ'.fintype : Fintype Γ' :=
⟨⟨{.blank, .bit true, .bit false, .bra, .ket, .comma}, by decide⟩,
by intro; cases_type* Γ' Bool <;> | Mathlib.Computability.Encoding.67_0.rxHMk1LMmv2SbJ8 | instance Γ'.fintype : Fintype Γ' | Mathlib_Computability_Encoding |
⊢ ∀ (n : PosNum), decodePosNum (encodePosNum n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | intro n | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
| Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
n : PosNum
⊢ decodePosNum (encodePosNum n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | induction' n with m hm m hm | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
| Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case one
⊢ decodePosNum (encodePosNum PosNum.one) = PosNum.one | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | unfold encodePosNum decodePosNum | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case bit1
m : PosNum
hm : decodePosNum (encodePosNum m) = m
⊢ decodePosNum (encodePosNum (PosNum.bit1 m)) = PosNum.bit1 m | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | unfold encodePosNum decodePosNum | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case bit0
m : PosNum
hm : decodePosNum (encodePosNum m) = m
⊢ decodePosNum (encodePosNum (PosNum.bit0 m)) = PosNum.bit0 m | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | unfold encodePosNum decodePosNum | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case one
⊢ (if [] = [] then PosNum.one else PosNum.bit1 (decodePosNum [])) = PosNum.one | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rfl | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> unfold encodePosNum decodePosNum
· | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case bit1
m : PosNum
hm : decodePosNum (encodePosNum m) = m
⊢ (if encodePosNum m = [] then PosNum.one else PosNum.bit1 (decodePosNum (encodePosNum m))) = PosNum.bit1 m | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [hm] | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> unfold encodePosNum decodePosNum
· rfl
· | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case bit1
m : PosNum
hm : decodePosNum (encodePosNum m) = m
⊢ (if encodePosNum m = [] then PosNum.one else PosNum.bit1 m) = PosNum.bit1 m | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | exact if_neg (encodePosNum_nonempty m) | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> unfold encodePosNum decodePosNum
· rfl
· rw [hm]
| Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
case bit0
m : PosNum
hm : decodePosNum (encodePosNum m) = m
⊢ PosNum.bit0 (decodePosNum (encodePosNum m)) = PosNum.bit0 m | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | exact congr_arg PosNum.bit0 hm | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by
intro n
induction' n with m hm m hm <;> unfold encodePosNum decodePosNum
· rfl
· rw [hm]
exact if_neg (encodePosNum_nonempty m)
· | Mathlib.Computability.Encoding.133_0.rxHMk1LMmv2SbJ8 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | Mathlib_Computability_Encoding |
⊢ ∀ (n : Num), decodeNum (encodeNum n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | intro n | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
| Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
n : Num
⊢ decodeNum (encodeNum n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | cases' n with n | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
| Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case zero
⊢ decodeNum (encodeNum Num.zero) = Num.zero | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | unfold encodeNum decodeNum | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> | Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case pos
n : PosNum
⊢ decodeNum (encodeNum (Num.pos n)) = Num.pos n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | unfold encodeNum decodeNum | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> | Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case zero
⊢ (if
(match Num.zero with
| Num.zero => []
| Num.pos n => encodePosNum n) =
[] then
Num.zero
else
↑(decodePosNum
(match Num.zero with
| Num.zero => []
| Num.pos n => encodePosNum n))) =
Num.zero | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rfl | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> unfold encodeNum decodeNum
· | Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case pos
n : PosNum
⊢ (if
(match Num.pos n with
| Num.zero => []
| Num.pos n => encodePosNum n) =
[] then
Num.zero
else
↑(decodePosNum
(match Num.pos n with
| Num.zero => []
| Num.pos n => encodePosNum n))) =
Num.pos n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [decode_encodePosNum n] | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
| Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case pos
n : PosNum
⊢ (if
(match Num.pos n with
| Num.zero => []
| Num.pos n => encodePosNum n) =
[] then
Num.zero
else ↑n) =
Num.pos n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [PosNum.cast_to_num] | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
rw [decode_encodePosNum n]
| Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
case pos
n : PosNum
⊢ (if
(match Num.pos n with
| Num.zero => []
| Num.pos n => encodePosNum n) =
[] then
Num.zero
else Num.pos n) =
Num.pos n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | exact if_neg (encodePosNum_nonempty n) | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
rw [decode_encodePosNum n]
rw [PosNum.cast_to_num]
| Mathlib.Computability.Encoding.142_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n | Mathlib_Computability_Encoding |
⊢ ∀ (n : ℕ), decodeNat (encodeNat n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | intro n | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
| Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
n : ℕ
⊢ decodeNat (encodeNat n) = n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | conv_rhs => rw [← Num.to_of_nat n] | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
intro n
| Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
n : ℕ
| n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [← Num.to_of_nat n] | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
intro n
conv_rhs => | Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
n : ℕ
| n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [← Num.to_of_nat n] | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
intro n
conv_rhs => | Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
n : ℕ
| n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [← Num.to_of_nat n] | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
intro n
conv_rhs => | Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
n : ℕ
⊢ decodeNat (encodeNat n) = ↑↑n | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | exact congr_arg ((↑) : Num → ℕ) (decode_encodeNum n) | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n := by
intro n
conv_rhs => rw [← Num.to_of_nat n]
| Mathlib.Computability.Encoding.151_0.rxHMk1LMmv2SbJ8 | theorem decode_encodeNat : ∀ n, decodeNat (encodeNat n) = n | Mathlib_Computability_Encoding |
x : ℕ
⊢ decodeNat (List.map sectionΓ'Bool ((fun x => List.map inclusionBoolΓ' (encodeNat x)) x)) = x | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [List.map_map, leftInverse_section_inclusion.id, List.map_id, decode_encodeNat] | /-- A binary encoding of ℕ in Γ'. -/
def encodingNatΓ' : Encoding ℕ where
Γ := Γ'
encode x := List.map inclusionBoolΓ' (encodeNat x)
decode x := some (decodeNat (List.map sectionΓ'Bool x))
decode_encode x :=
congr_arg _ <| by
-- Porting note: `rw` can't unify `g ∘ f` with `fun x => g (f x)`, used `Lef... | Mathlib.Computability.Encoding.170_0.rxHMk1LMmv2SbJ8 | /-- A binary encoding of ℕ in Γ'. -/
def encodingNatΓ' : Encoding ℕ where
Γ | Mathlib_Computability_Encoding |
α : Type u
e : Encoding α
inst✝ : Encodable e.Γ
⊢ #α ≤ ℵ₀ | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _) | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ := by
| Mathlib.Computability.Encoding.247_0.rxHMk1LMmv2SbJ8 | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ | Mathlib_Computability_Encoding |
α : Type u
e : Encoding α
inst✝ : Encodable e.Γ
⊢ lift.{u, v} #(List e.Γ) ≤ lift.{v, u} ℵ₀ | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | simp only [Cardinal.lift_aleph0, Cardinal.lift_le_aleph0] | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ := by
refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _)
| Mathlib.Computability.Encoding.247_0.rxHMk1LMmv2SbJ8 | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ | Mathlib_Computability_Encoding |
α : Type u
e : Encoding α
inst✝ : Encodable e.Γ
⊢ #(List e.Γ) ≤ ℵ₀ | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | cases' isEmpty_or_nonempty e.Γ with h h | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ := by
refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _)
simp only [Cardinal.lift_aleph0, Cardinal.lift_le_aleph0]
| Mathlib.Computability.Encoding.247_0.rxHMk1LMmv2SbJ8 | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ | Mathlib_Computability_Encoding |
case inl
α : Type u
e : Encoding α
inst✝ : Encodable e.Γ
h : IsEmpty e.Γ
⊢ #(List e.Γ) ≤ ℵ₀ | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | simp only [Cardinal.mk_le_aleph0] | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ := by
refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _)
simp only [Cardinal.lift_aleph0, Cardinal.lift_le_aleph0]
cases' isEmpty_or_nonempty e.Γ with h h
· | Mathlib.Computability.Encoding.247_0.rxHMk1LMmv2SbJ8 | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ | Mathlib_Computability_Encoding |
case inr
α : Type u
e : Encoding α
inst✝ : Encodable e.Γ
h : Nonempty e.Γ
⊢ #(List e.Γ) ≤ ℵ₀ | /-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import computability.encodi... | rw [Cardinal.mk_list_eq_aleph0] | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ := by
refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _)
simp only [Cardinal.lift_aleph0, Cardinal.lift_le_aleph0]
cases' isEmpty_or_nonempty e.Γ with h h
· simp only [Cardinal.mk_le_aleph0]
· | Mathlib.Computability.Encoding.247_0.rxHMk1LMmv2SbJ8 | theorem Encoding.card_le_aleph0 {α : Type u} (e : Encoding.{u, v} α) [Encodable e.Γ] :
#α ≤ ℵ₀ | Mathlib_Computability_Encoding |
ι : Type u_1
c : ComplexShape ι
⊢ symm (symm c) = c | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | ext | @[simp]
theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c := by
| Mathlib.Algebra.Homology.ComplexShape.99_0.XSrMOWOP54vJcCl | @[simp]
theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c | Mathlib_Algebra_Homology_ComplexShape |
case Rel.h.h.a
ι : Type u_1
c : ComplexShape ι
x✝¹ x✝ : ι
⊢ Rel (symm (symm c)) x✝¹ x✝ ↔ Rel c x✝¹ x✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | simp | @[simp]
theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c := by
ext
| Mathlib.Algebra.Homology.ComplexShape.99_0.XSrMOWOP54vJcCl | @[simp]
theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ j✝ j'✝ : ι
w : Relation.Comp c₁.Rel c₂.Rel i✝ j✝
w' : Relation.Comp c₁.Rel c₂.Rel i✝ j'✝
⊢ j✝ = j'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | obtain ⟨k, w₁, w₂⟩ := w | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
| Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ j✝ j'✝ : ι
w' : Relation.Comp c₁.Rel c₂.Rel i✝ j'✝
k : ι
w₁ : Rel c₁ i✝ k
w₂ : Rel c₂ k j✝
⊢ j✝ = j'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | obtain ⟨k', w₁', w₂'⟩ := w' | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
| Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro.intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ j✝ j'✝ k : ι
w₁ : Rel c₁ i✝ k
w₂ : Rel c₂ k j✝
k' : ι
w₁' : Rel c₁ i✝ k'
w₂' : Rel c₂ k' j'✝
⊢ j✝ = j'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rw [c₁.next_eq w₁ w₁'] at w₂ | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
| Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro.intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ j✝ j'✝ k : ι
w₁ : Rel c₁ i✝ k
k' : ι
w₂ : Rel c₂ k' j✝
w₁' : Rel c₁ i✝ k'
w₂' : Rel c₂ k' j'✝
⊢ j✝ = j'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact c₂.next_eq w₂ w₂' | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
rw [c₁.next_eq w₁ w₁'] at w₂
... | Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ i'✝ j✝ : ι
w : Relation.Comp c₁.Rel c₂.Rel i✝ j✝
w' : Relation.Comp c₁.Rel c₂.Rel i'✝ j✝
⊢ i✝ = i'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | obtain ⟨k, w₁, w₂⟩ := w | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
rw [c₁.next_eq w₁ w₁'] at w₂
... | Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ i'✝ j✝ : ι
w' : Relation.Comp c₁.Rel c₂.Rel i'✝ j✝
k : ι
w₁ : Rel c₁ i✝ k
w₂ : Rel c₂ k j✝
⊢ i✝ = i'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | obtain ⟨k', w₁', w₂'⟩ := w' | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
rw [c₁.next_eq w₁ w₁'] at w₂
... | Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro.intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ i'✝ j✝ k : ι
w₁ : Rel c₁ i✝ k
w₂ : Rel c₂ k j✝
k' : ι
w₁' : Rel c₁ i'✝ k'
w₂' : Rel c₂ k' j✝
⊢ i✝ = i'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rw [c₂.prev_eq w₂ w₂'] at w₁ | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
rw [c₁.next_eq w₁ w₁'] at w₂
... | Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
case intro.intro.intro.intro
ι : Type u_1
c₁ c₂ : ComplexShape ι
i✝ i'✝ j✝ k : ι
w₂ : Rel c₂ k j✝
k' : ι
w₁ : Rel c₁ i✝ k'
w₁' : Rel c₁ i'✝ k'
w₂' : Rel c₂ k' j✝
⊢ i✝ = i'✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact c₁.prev_eq w₁ w₁' | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel := Relation.Comp c₁.Rel c₂.Rel
next_eq w w' := by
obtain ⟨k, w₁, w₂⟩ := w
obtain ⟨k', w₁', w₂'⟩ := w'
rw [c₁.next_eq w₁ w₁'] at w₂
... | Mathlib.Algebra.Homology.ComplexShape.109_0.XSrMOWOP54vJcCl | /-- The "composition" of two `ComplexShape`s.
We need this to define "related in k steps" later.
-/
@[simp]
def trans (c₁ c₂ : ComplexShape ι) : ComplexShape ι where
Rel | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i : ι
⊢ Subsingleton { j // Rel c i j } | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | constructor | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } := by
| Mathlib.Algebra.Homology.ComplexShape.128_0.XSrMOWOP54vJcCl | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq
ι : Type u_1
c : ComplexShape ι
i : ι
⊢ ∀ (a b : { j // Rel c i j }), a = b | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rintro ⟨j, rij⟩ ⟨k, rik⟩ | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } := by
constructor
| Mathlib.Algebra.Homology.ComplexShape.128_0.XSrMOWOP54vJcCl | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq.mk.mk
ι : Type u_1
c : ComplexShape ι
i j : ι
rij : Rel c i j
k : ι
rik : Rel c i k
⊢ { val := j, property := rij } = { val := k, property := rik } | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | congr | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } := by
constructor
rintro ⟨j, rij⟩ ⟨k, rik⟩
| Mathlib.Algebra.Homology.ComplexShape.128_0.XSrMOWOP54vJcCl | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq.mk.mk.e_val
ι : Type u_1
c : ComplexShape ι
i j : ι
rij : Rel c i j
k : ι
rik : Rel c i k
⊢ j = k | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact c.next_eq rij rik | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } := by
constructor
rintro ⟨j, rij⟩ ⟨k, rik⟩
congr
| Mathlib.Algebra.Homology.ComplexShape.128_0.XSrMOWOP54vJcCl | instance subsingleton_next (c : ComplexShape ι) (i : ι) : Subsingleton { j // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
j : ι
⊢ Subsingleton { i // Rel c i j } | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | constructor | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } := by
| Mathlib.Algebra.Homology.ComplexShape.134_0.XSrMOWOP54vJcCl | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq
ι : Type u_1
c : ComplexShape ι
j : ι
⊢ ∀ (a b : { i // Rel c i j }), a = b | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rintro ⟨i, rik⟩ ⟨j, rjk⟩ | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } := by
constructor
| Mathlib.Algebra.Homology.ComplexShape.134_0.XSrMOWOP54vJcCl | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq.mk.mk
ι : Type u_1
c : ComplexShape ι
j✝ i : ι
rik : Rel c i j✝
j : ι
rjk : Rel c j j✝
⊢ { val := i, property := rik } = { val := j, property := rjk } | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | congr | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } := by
constructor
rintro ⟨i, rik⟩ ⟨j, rjk⟩
| Mathlib.Algebra.Homology.ComplexShape.134_0.XSrMOWOP54vJcCl | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
case allEq.mk.mk.e_val
ι : Type u_1
c : ComplexShape ι
j✝ i : ι
rik : Rel c i j✝
j : ι
rjk : Rel c j j✝
⊢ i = j | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact c.prev_eq rik rjk | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } := by
constructor
rintro ⟨i, rik⟩ ⟨j, rjk⟩
congr
| Mathlib.Algebra.Homology.ComplexShape.134_0.XSrMOWOP54vJcCl | instance subsingleton_prev (c : ComplexShape ι) (j : ι) : Subsingleton { i // c.Rel i j } | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ next c i = j | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | apply c.next_eq _ h | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j := by
| Mathlib.Algebra.Homology.ComplexShape.154_0.XSrMOWOP54vJcCl | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ Rel c i (next c i) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rw [next] | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j := by
apply c.next_eq _ h
| Mathlib.Algebra.Homology.ComplexShape.154_0.XSrMOWOP54vJcCl | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ Rel c i (if h : ∃ j, Rel c i j then Exists.choose h else i) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rw [dif_pos] | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j := by
apply c.next_eq _ h
rw [next]
| Mathlib.Algebra.Homology.ComplexShape.154_0.XSrMOWOP54vJcCl | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ Rel c i (Exists.choose ?hc)
case hc ι : Type u_1 c : ComplexShape ι i j : ι h : Rel c i j ⊢ ∃ j, Rel c i j | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact Exists.choose_spec ⟨j, h⟩ | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j := by
apply c.next_eq _ h
rw [next]
rw [dif_pos]
| Mathlib.Algebra.Homology.ComplexShape.154_0.XSrMOWOP54vJcCl | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ prev c j = i | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | apply c.prev_eq _ h | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i := by
| Mathlib.Algebra.Homology.ComplexShape.161_0.XSrMOWOP54vJcCl | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ Rel c (prev c j) j | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | rw [prev, dif_pos] | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i := by
apply c.prev_eq _ h
| Mathlib.Algebra.Homology.ComplexShape.161_0.XSrMOWOP54vJcCl | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i | Mathlib_Algebra_Homology_ComplexShape |
ι : Type u_1
c : ComplexShape ι
i j : ι
h : Rel c i j
⊢ Rel c (Exists.choose ?hc) j
case hc ι : Type u_1 c : ComplexShape ι i j : ι h : Rel c i j ⊢ ∃ i, Rel c i j | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | exact Exists.choose_spec (⟨i, h⟩ : ∃ k, c.Rel k j) | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i := by
apply c.prev_eq _ h
rw [prev, dif_pos]
| Mathlib.Algebra.Homology.ComplexShape.161_0.XSrMOWOP54vJcCl | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i | Mathlib_Algebra_Homology_ComplexShape |
α : Type u_1
x : ℤˣ
⊢ x ∈ {1, -1} | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | cases Int.units_eq_one_or x | instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by | Mathlib.Data.Fintype.Units.20_0.6sF1mNVGQq4PLsW | instance UnitsInt.fintype : Fintype ℤˣ | Mathlib_Data_Fintype_Units |
case inl
α : Type u_1
x : ℤˣ
h✝ : x = 1
⊢ x ∈ {1, -1} | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | simp [*] | instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int.units_eq_one_or x <;> | Mathlib.Data.Fintype.Units.20_0.6sF1mNVGQq4PLsW | instance UnitsInt.fintype : Fintype ℤˣ | Mathlib_Data_Fintype_Units |
case inr
α : Type u_1
x : ℤˣ
h✝ : x = -1
⊢ x ∈ {1, -1} | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | simp [*] | instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int.units_eq_one_or x <;> | Mathlib.Data.Fintype.Units.20_0.6sF1mNVGQq4PLsW | instance UnitsInt.fintype : Fintype ℤˣ | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝² : GroupWithZero α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ card α = card αˣ + 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | rw [eq_comm, Fintype.card_congr (unitsEquivNeZero α)] | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 := by
| Mathlib.Data.Fintype.Units.37_0.6sF1mNVGQq4PLsW | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝² : GroupWithZero α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ card { a // a ≠ 0 } + 1 = card α | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | have := Fintype.card_congr (Equiv.sumCompl (· = (0 : α))) | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 := by
rw [eq_comm, Fintype.card_congr (unitsEquivNeZero α)]
| Mathlib.Data.Fintype.Units.37_0.6sF1mNVGQq4PLsW | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝² : GroupWithZero α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
this : card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) = card α
⊢ card { a // a ≠ 0 } + 1 = card α | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | rwa [Fintype.card_sum, add_comm, Fintype.card_subtype_eq] at this | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 := by
rw [eq_comm, Fintype.card_congr (unitsEquivNeZero α)]
have := Fintype.card_congr (Equiv.sumCompl (· = (0 : α)))
| Mathlib.Data.Fintype.Units.37_0.6sF1mNVGQq4PLsW | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝¹ : GroupWithZero α
inst✝ : Finite α
⊢ Nat.card α = Nat.card αˣ + 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | have : Fintype α := Fintype.ofFinite α | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 := by
| Mathlib.Data.Fintype.Units.43_0.6sF1mNVGQq4PLsW | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝¹ : GroupWithZero α
inst✝ : Finite α
this : Fintype α
⊢ Nat.card α = Nat.card αˣ + 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | classical
rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, Fintype.card_eq_card_units_add_one] | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 := by
have : Fintype α := Fintype.ofFinite α
| Mathlib.Data.Fintype.Units.43_0.6sF1mNVGQq4PLsW | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝¹ : GroupWithZero α
inst✝ : Finite α
this : Fintype α
⊢ Nat.card α = Nat.card αˣ + 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, Fintype.card_eq_card_units_add_one] | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 := by
have : Fintype α := Fintype.ofFinite α
classical
| Mathlib.Data.Fintype.Units.43_0.6sF1mNVGQq4PLsW | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 | Mathlib_Data_Fintype_Units |
α : Type u_1
inst✝² : GroupWithZero α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ card αˣ = card α - 1 | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Units
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.un... | rw [@Fintype.card_eq_card_units_add_one α, Nat.add_sub_cancel] | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 := by
| Mathlib.Data.Fintype.Units.49_0.6sF1mNVGQq4PLsW | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 | Mathlib_Data_Fintype_Units |
θ : ℂ
⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub]
congr 3; ring_nf | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub] | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ cexp (θ * I - -θ * I) = -1 ↔ cexp (2 * θ * I) = -1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | congr 3 | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ cexp (θ * I - -θ * I) = -1 ↔ cexp (2 * θ * I) = -1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | ring_nf | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [cos, h, ← exp_pi_mul_I, exp_eq_exp_iff_exists_int, mul_right_comm] | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
⊢ (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | refine' exists_congr fun x => _ | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
x : ℤ
⊢ 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | refine' (iff_of_eq <| congr_arg _ _).trans (mul_right_inj' <| mul_ne_zero two_ne_zero I_ne_zero) | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
x : ℤ
⊢ ↑π * I + ↑x * (2 * ↑π * I) = 2 * I * ((2 * ↑x + 1) * ↑π / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | field_simp | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
x : ℤ
⊢ (↑π * I + ↑x * (2 * ↑π * I)) * 2 = 2 * I * ((2 * ↑x + 1) * ↑π) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | ring | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1 := by
rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _)... | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.31_0.wRglntQQQHH0e1R | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ cos θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ (2 * ↑k + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [← not_exists, not_iff_not, cos_eq_zero_iff] | theorem cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.42_0.wRglntQQQHH0e1R | theorem cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ sin θ = 0 ↔ ∃ k, θ = ↑k * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff] | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) ↔ ∃ k, θ = ↑k * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | constructor | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case mp
θ : ℂ
⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) → ∃ k, θ = ↑k * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rintro ⟨k, hk⟩ | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case mp.intro
θ : ℂ
k : ℤ
hk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2
⊢ ∃ k, θ = ↑k * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | use k + 1 | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case h
θ : ℂ
k : ℤ
hk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2
⊢ θ = ↑(k + 1) * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | field_simp [eq_add_of_sub_eq hk] | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case h
θ : ℂ
k : ℤ
hk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2
⊢ (2 * ↑k + 1) * ↑π + ↑π = (↑k + 1) * ↑π * 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | ring | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
field_simp [eq_add_of_sub_eq hk]
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case mpr
θ : ℂ
⊢ (∃ k, θ = ↑k * ↑π) → ∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rintro ⟨k, rfl⟩ | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
field_simp [eq_add_of_sub_eq hk]
ring
· | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case mpr.intro
k : ℤ
⊢ ∃ k_1, ↑k * ↑π - ↑π / 2 = (2 * ↑k_1 + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | use k - 1 | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
field_simp [eq_add_of_sub_eq hk]
ring
· rintro ⟨k, rfl⟩
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case h
k : ℤ
⊢ ↑k * ↑π - ↑π / 2 = (2 * ↑(k - 1) + 1) * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | field_simp | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
field_simp [eq_add_of_sub_eq hk]
ring
· rintro ⟨k, rfl⟩
use k - 1
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
case h
k : ℤ
⊢ ↑k * ↑π * 2 - ↑π = (2 * (↑k - 1) + 1) * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | ring | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := by
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
constructor
· rintro ⟨k, hk⟩
use k + 1
field_simp [eq_add_of_sub_eq hk]
ring
· rintro ⟨k, rfl⟩
use k - 1
field_simp
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.46_0.wRglntQQQHH0e1R | theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ sin θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ ↑k * ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [← not_exists, not_iff_not, sin_eq_zero_iff] | theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.59_0.wRglntQQQHH0e1R | theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ tan θ = 0 ↔ ∃ k, θ = ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | have h := (sin_two_mul θ).symm | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.63_0.wRglntQQQHH0e1R | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : 2 * sin θ * cos θ = sin (2 * θ)
⊢ tan θ = 0 ↔ ∃ k, θ = ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [mul_assoc] at h | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := by
have h := (sin_two_mul θ).symm
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.63_0.wRglntQQQHH0e1R | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : 2 * (sin θ * cos θ) = sin (2 * θ)
⊢ tan θ = 0 ↔ ∃ k, θ = ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← zero_mul (1 / 2 : ℂ), mul_one_div,
CancelDenoms.cancel_factors_eq_div h two_ne_zero, mul_comm] | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := by
have h := (sin_two_mul θ).symm
rw [mul_assoc] at h
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.63_0.wRglntQQQHH0e1R | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
h : 2 * (sin θ * cos θ) = sin (2 * θ)
⊢ sin (θ * 2) / 2 = 0 / 2 ↔ ∃ k, θ = ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | simpa only [zero_div, zero_mul, Ne.def, not_false_iff, field_simps] using
sin_eq_zero_iff | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := by
have h := (sin_two_mul θ).symm
rw [mul_assoc] at h
rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← zero_mul (1 / 2 : ℂ), mul_one_div,
CancelDenoms.cancel_factors_eq_div h two_ne_zero, mul_comm]
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.63_0.wRglntQQQHH0e1R | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
θ : ℂ
⊢ tan θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [← not_exists, not_iff_not, tan_eq_zero_iff] | theorem tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 := by
| Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.72_0.wRglntQQQHH0e1R | theorem tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
n : ℤ
⊢ ∃ k, ↑n * ↑π / 2 = ↑k * ↑π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | use n | theorem tan_int_mul_pi_div_two (n : ℤ) : tan (n * π / 2) = 0 :=
tan_eq_zero_iff.mpr (by | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.76_0.wRglntQQQHH0e1R | theorem tan_int_mul_pi_div_two (n : ℤ) : tan (n * π / 2) = 0 | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
x y : ℂ
⊢ cos x - cos y = 0 ↔ -2 * sin ((x + y) / 2) * sin ((x - y) / 2) = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#a... | rw [cos_sub_cos] | theorem cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x :=
calc
cos x = cos y ↔ cos x - cos y = 0 := sub_eq_zero.symm
_ ↔ -2 * sin ((x + y) / 2) * sin ((x - y) / 2) = 0 := by | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex.80_0.wRglntQQQHH0e1R | theorem cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x | Mathlib_Analysis_SpecialFunctions_Trigonometric_Complex |
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