feat: Associativity of multiplication
This commit is contained in:
jstoobysmith
parent
bc2db84389
commit
99ccbb5d04
3 changed files with 451 additions and 197 deletions
|
|
@ -166,7 +166,7 @@ lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
|
|||
rw [← Equiv.symm_apply_eq]
|
||||
funext y
|
||||
rw [indexValueIso_symm_apply', indexValueIso_symm_apply']
|
||||
simp [colorsIndexCast]
|
||||
simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.apply_symm_apply]
|
||||
|
||||
|
|
@ -396,150 +396,226 @@ def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃ (X ⊕ Fin 1)
|
|||
def unmarkFirst {X : Type} : Marked d X n.succ ≃ Marked d (X ⊕ Fin 1) n :=
|
||||
mapIso d (unmarkFirstSet X n)
|
||||
|
||||
@[simps!]
|
||||
def notInImage {n m : ℕ} (f : Fin m ↪ Fin (n + m)) :
|
||||
Fin n ≃ {x // x ∈ (Finset.image (⇑f) Finset.univ)ᶜ} :=
|
||||
(Finset.orderIsoOfFin (Finset.image f Finset.univ)ᶜ (by rw [Finset.card_compl,
|
||||
Finset.card_image_of_injective _ (Function.Embedding.injective f)]; simp)).toEquiv
|
||||
/-!
|
||||
|
||||
@[simps!]
|
||||
def splitMarkedIndices {n m : ℕ} (f : Fin m ↪ Fin (n + m)) : Fin (n + m) ≃ Fin n ⊕ Fin m :=
|
||||
## Marking elements.
|
||||
|
||||
-/
|
||||
section markingElements
|
||||
|
||||
variable [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
|
||||
/-- Splits a type based on an embedding from `Fin n` into elements not in the image of the
|
||||
embedding, and elements in the image. -/
|
||||
def markEmbeddingSet {n : ℕ} (f : Fin n ↪ X) :
|
||||
X ≃ {x // x ∈ (Finset.image f Finset.univ)ᶜ} ⊕ Fin n :=
|
||||
(Equiv.Set.sumCompl (Set.range ⇑f)).symm.trans <|
|
||||
(Equiv.sumComm _ _).trans <|
|
||||
Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp)).trans (notInImage f).symm) <|
|
||||
Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp))) <|
|
||||
(Function.Embedding.toEquivRange f).symm
|
||||
|
||||
lemma markEmbeddingSet_on_mem {n : ℕ} (f : Fin n ↪ X) (x : X)
|
||||
(hx : x ∈ Finset.image f Finset.univ) : markEmbeddingSet f x =
|
||||
Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩) := by
|
||||
rw [markEmbeddingSet]
|
||||
simp only [Equiv.trans_apply, Equiv.sumComm_apply, Equiv.sumCongr_apply]
|
||||
rw [Equiv.Set.sumCompl_symm_apply_of_mem]
|
||||
rfl
|
||||
|
||||
lemma markEmbeddingSet_on_not_mem {n : ℕ} (f : Fin n ↪ X) (x : X)
|
||||
(hx : ¬ x ∈ (Finset.image f Finset.univ)) : markEmbeddingSet f x =
|
||||
Sum.inl ⟨x, by simpa using hx⟩ := by
|
||||
rw [markEmbeddingSet]
|
||||
simp only [Equiv.trans_apply, Equiv.sumComm_apply, Equiv.sumCongr_apply]
|
||||
rw [Equiv.Set.sumCompl_symm_apply_of_not_mem]
|
||||
rfl
|
||||
simpa using hx
|
||||
|
||||
/-- Marks the indicies of tensor in the image of an embedding. -/
|
||||
@[simps!]
|
||||
def oneEmbed {n : ℕ} (i : Fin n.succ) : Fin 1 ↪ Fin n.succ :=
|
||||
⟨fun _ => i, fun x y => by fin_cases x; fin_cases y; simp⟩
|
||||
def markEmbedding {n : ℕ} (f : Fin n ↪ X) :
|
||||
RealLorentzTensor d X ≃ Marked d {x // x ∈ (Finset.image f Finset.univ)ᶜ} n :=
|
||||
mapIso d (markEmbeddingSet f)
|
||||
|
||||
@[simps!]
|
||||
def splitMarkedIndicesOne {n : ℕ} (i : Fin n.succ) : Fin n.succ ≃ Fin n ⊕ Fin 1 :=
|
||||
splitMarkedIndices (oneEmbed i)
|
||||
|
||||
lemma splitMarkedIndicesOne_self {n : ℕ} (i : Fin n.succ) :
|
||||
splitMarkedIndicesOne i i = Sum.inr 0 := by
|
||||
rw [splitMarkedIndicesOne, splitMarkedIndices]
|
||||
have h1 : (Equiv.Set.sumCompl (Set.range (oneEmbed i))).symm i =
|
||||
Sum.inl ⟨i, by simp⟩ := by
|
||||
rw [Equiv.Set.sumCompl_symm_apply_of_mem]
|
||||
simp
|
||||
rw [h1]
|
||||
change Sum.inr ((⇑(oneEmbed i).toEquivRange.symm) (⟨(oneEmbed i) 0, _⟩)) = Sum.inr 0
|
||||
congr
|
||||
rw [Function.Embedding.toEquivRange_symm_apply_self]
|
||||
|
||||
lemma notInImage_oneEmbed {n : ℕ} (i : Fin n.succ) :
|
||||
(notInImage (oneEmbed i)).toFun = fun x => ⟨i.succAbove x, by simpa using Fin.succAbove_ne i x⟩ := by
|
||||
lemma markEmbeddingSet_on_mem_indexValue_apply {n : ℕ} (f : Fin n ↪ X) (T : RealLorentzTensor d X)
|
||||
(i : IndexValue d (markEmbedding f T).color) (x : X) (hx : x ∈ (Finset.image f Finset.univ)) :
|
||||
i (markEmbeddingSet f x) = colorsIndexCast (congrArg ((markEmbedding f) T).color
|
||||
(markEmbeddingSet_on_mem f x hx).symm)
|
||||
(i (Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩))) := by
|
||||
simp [colorsIndexCast]
|
||||
symm
|
||||
simp [notInImage]
|
||||
funext x
|
||||
apply Subtype.ext
|
||||
rw [Finset.coe_orderIsoOfFin_apply]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [markEmbeddingSet_on_mem f x hx]
|
||||
|
||||
lemma markEmbeddingSet_on_not_mem_indexValue_apply {n : ℕ}
|
||||
(f : Fin n ↪ X) (T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color)
|
||||
(x : X) (hx : ¬ x ∈ (Finset.image f Finset.univ)) :
|
||||
i (markEmbeddingSet f x) = colorsIndexCast (congrArg ((markEmbedding f) T).color
|
||||
(markEmbeddingSet_on_not_mem f x hx).symm) (i (Sum.inl ⟨x, by simpa using hx⟩)) := by
|
||||
simp [colorsIndexCast]
|
||||
symm
|
||||
apply congrFun
|
||||
symm
|
||||
apply Finset.orderEmbOfFin_unique
|
||||
intro x
|
||||
simp
|
||||
exact Fin.succAbove_ne i x
|
||||
exact Fin.strictMono_succAbove i
|
||||
|
||||
lemma splitMarkedIndicesOne_not_self {n : ℕ} {i j : Fin n.succ.succ} (hij : i ≠ j) :
|
||||
splitMarkedIndicesOne i j = Sum.inl ((Fin.predAbove 0 i).predAbove j) := by
|
||||
rw [splitMarkedIndicesOne, splitMarkedIndices]
|
||||
have h1 : (Equiv.Set.sumCompl (Set.range (oneEmbed i))).symm j =
|
||||
Sum.inr ⟨j, by simp [hij]⟩ := by
|
||||
rw [Equiv.Set.sumCompl_symm_apply_of_not_mem]
|
||||
simp
|
||||
rw [h1]
|
||||
change Sum.inl ((⇑(notInImage (oneEmbed i)).symm ∘ ⇑(Equiv.subtypeEquivRight _))
|
||||
⟨j, _⟩) = _
|
||||
apply congrArg
|
||||
simp
|
||||
rw [Equiv.symm_apply_eq]
|
||||
change _ = (notInImage (oneEmbed i)).toFun ((Fin.predAbove 0 i).predAbove j)
|
||||
rw [notInImage_oneEmbed]
|
||||
simp
|
||||
apply Subtype.ext
|
||||
simp
|
||||
simp [Fin.succAbove, Fin.predAbove, Fin.lt_def]
|
||||
split <;> split <;> simp <;> split <;> apply Fin.ext_iff.mpr
|
||||
<;> rename_i _ h2 h3 h4 <;> simp at h1 h2 h3 h4 <;> omega
|
||||
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [markEmbeddingSet_on_not_mem f x hx]
|
||||
|
||||
/-- An equivalence between the IndexValues for a tensor `T` and the corresponding
|
||||
tensor with indicies maked by an embedding. -/
|
||||
@[simps!]
|
||||
def markSelectedIndex (i : Fin n.succ) : Marked d X n.succ ≃ Marked d (X ⊕ Fin n) 1 :=
|
||||
mapIso d ((Equiv.sumCongr (Equiv.refl X) (splitMarkedIndicesOne i)).trans
|
||||
(Equiv.sumAssoc _ _ _).symm)
|
||||
def markEmbeddingIndexValue {n : ℕ} (f : Fin n ↪ X) (T : RealLorentzTensor d X) :
|
||||
IndexValue d T.color ≃ IndexValue d (markEmbedding f T).color :=
|
||||
indexValueIso d (markEmbeddingSet f) (
|
||||
(Equiv.comp_symm_eq (markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl)
|
||||
|
||||
lemma markSelectedIndex_markedColor (T : Marked d X n.succ) (i : Fin n.succ) :
|
||||
(markSelectedIndex i T).markedColor 0 = T.markedColor i := rfl
|
||||
lemma markEmbeddingIndexValue_apply_symm_on_mem {n : ℕ}
|
||||
(f : Fin n.succ ↪ X) (T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color)
|
||||
(x : X) (hx : x ∈ (Finset.image f Finset.univ)) :
|
||||
(markEmbeddingIndexValue f T).symm i x = (colorsIndexCast ((congrFun ((Equiv.comp_symm_eq
|
||||
(markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl) x).trans
|
||||
(congrArg ((markEmbedding f) T).color (markEmbeddingSet_on_mem f x hx)))).symm
|
||||
(i (Sum.inr (f.toEquivRange.symm ⟨x, by simpa using hx⟩))) := by
|
||||
rw [markEmbeddingIndexValue, indexValueIso_symm_apply']
|
||||
rw [markEmbeddingSet_on_mem_indexValue_apply f T i x hx]
|
||||
simp [colorsIndexCast]
|
||||
|
||||
/-! TODO: Simplify prove and move. -/
|
||||
lemma succAbove_predAbove_predAbove (i p : Fin n.succ.succ) (hip : i ≠ p) :
|
||||
i.succAbove ((Fin.predAbove 0 i).predAbove p) = p := by
|
||||
simp [Fin.succAbove, Fin.predAbove, Fin.lt_def]
|
||||
split <;> split <;> simp <;> rw [Fin.ext_iff] <;> simp
|
||||
intro h
|
||||
all_goals
|
||||
rename_i h1 h2
|
||||
simp at h1 h2
|
||||
omega
|
||||
lemma markEmbeddingIndexValue_apply_symm_on_not_mem {n : ℕ} (f : Fin n.succ ↪ X)
|
||||
(T : RealLorentzTensor d X) (i : IndexValue d (markEmbedding f T).color) (x : X)
|
||||
(hx : ¬ x ∈ (Finset.image f Finset.univ)) : (markEmbeddingIndexValue f T).symm i x =
|
||||
(colorsIndexCast ((congrFun ((Equiv.comp_symm_eq
|
||||
(markEmbeddingSet f) ((markEmbedding f) T).color T.color).mp rfl) x).trans
|
||||
((congrArg ((markEmbedding f) T).color (markEmbeddingSet_on_not_mem f x hx))))).symm
|
||||
(i (Sum.inl ⟨x, by simpa using hx⟩)) := by
|
||||
rw [markEmbeddingIndexValue, indexValueIso_symm_apply']
|
||||
rw [markEmbeddingSet_on_not_mem_indexValue_apply f T i x hx]
|
||||
simp only [Nat.succ_eq_add_one, Function.comp_apply, markEmbedding_apply_color, colorsIndexCast,
|
||||
Equiv.cast_symm, id_eq, eq_mp_eq_cast, eq_mpr_eq_cast, Equiv.cast_apply, cast_cast, cast_eq,
|
||||
Equiv.cast_refl, Equiv.refl_symm]
|
||||
rfl
|
||||
|
||||
/-- Given an equivalence of types, an embedding `f` to an embedding `g`, the equivalence
|
||||
taking the complement of the image of `f` to the complement of the image of `g`. -/
|
||||
@[simps!]
|
||||
def equivDropPoint (e : Fin n.succ ≃ Fin m.succ) {i : Fin n.succ} {j : Fin m.succ} (he : e i = j) :
|
||||
Fin n ≃ Fin m :=
|
||||
(notInImage (oneEmbed i)).trans <|
|
||||
(Equiv.subtypeEquivOfSubtype' e).trans <|
|
||||
(Equiv.subtypeEquivRight (fun x => by simp [oneEmbed, Equiv.symm_apply_eq, he])).trans <|
|
||||
(notInImage (oneEmbed j)).symm
|
||||
def equivEmbedCompl (e : X ≃ Y) {f : Fin n ↪ X} {g : Fin n ↪ Y} (he : f.trans e = g) :
|
||||
{x // x ∈ (Finset.image f Finset.univ)ᶜ} ≃ {y // y ∈ (Finset.image g Finset.univ)ᶜ} :=
|
||||
(Equiv.subtypeEquivOfSubtype' e).trans <|
|
||||
(Equiv.subtypeEquivRight (fun x => by simp [← he, Equiv.eq_symm_apply]))
|
||||
|
||||
|
||||
lemma markSelectedIndex_mapIso (f : X ≃ Y) (e : Fin n.succ.succ ≃ Fin m.succ.succ) (i : Fin n.succ.succ)
|
||||
(j : Fin m.succ.succ) (he : e i = j) (T : Marked d X n.succ.succ) :
|
||||
mapIso d (Equiv.sumCongr (Equiv.sumCongr f (equivDropPoint e he)) (Equiv.refl (Fin 1)))
|
||||
(markSelectedIndex i T) = markSelectedIndex j (mapIso d (Equiv.sumCongr f e) T) := by
|
||||
rw [markSelectedIndex, markSelectedIndex]
|
||||
lemma markEmbedding_mapIso_right (e : X ≃ Y) (f : Fin n ↪ X) (g : Fin n ↪ Y) (he : f.trans e = g)
|
||||
(T : RealLorentzTensor d X) : markEmbedding g (mapIso d e T) =
|
||||
mapIso d (Equiv.sumCongr (equivEmbedCompl e he) (Equiv.refl (Fin n))) (markEmbedding f T) := by
|
||||
rw [markEmbedding, markEmbedding]
|
||||
erw [← Equiv.trans_apply, ← Equiv.trans_apply]
|
||||
rw [mapIso_trans, mapIso_trans]
|
||||
apply congrFun
|
||||
apply congrArg
|
||||
apply congrArg
|
||||
apply Equiv.ext
|
||||
intro x
|
||||
match x with
|
||||
| Sum.inl x => rfl
|
||||
| Sum.inr x =>
|
||||
simp only [Nat.succ_eq_add_one, Equiv.trans_apply, Equiv.coe_refl,
|
||||
Fin.isValue]
|
||||
change ((f.sumCongr (equivDropPoint e he)).sumCongr (Equiv.refl (Fin 1)))
|
||||
((Equiv.sumAssoc X (Fin n.succ) (Fin 1)).symm (Sum.inr ((splitMarkedIndicesOne i) x))) =
|
||||
(Equiv.sumAssoc Y (Fin m.succ) (Fin 1)).symm (Sum.inr ((splitMarkedIndicesOne j) (e x)))
|
||||
by_cases h : x = i
|
||||
subst h
|
||||
rw [he]
|
||||
rw [splitMarkedIndicesOne_self, splitMarkedIndicesOne_self]
|
||||
simp
|
||||
repeat apply congrArg
|
||||
refine Equiv.ext (fun x => ?_)
|
||||
simp only [Equiv.trans_apply, Equiv.sumCongr_apply, Equiv.coe_refl]
|
||||
by_cases hx : x ∈ Finset.image f Finset.univ
|
||||
· rw [markEmbeddingSet_on_mem f x hx, markEmbeddingSet_on_mem g (e x) (by simpa [← he] using hx)]
|
||||
subst he
|
||||
rw [splitMarkedIndicesOne_not_self (fun a => h a.symm), splitMarkedIndicesOne_not_self]
|
||||
simp [equivDropPoint]
|
||||
rw [Equiv.symm_apply_eq]
|
||||
change _ =
|
||||
(notInImage (oneEmbed (e i))).toFun ((Fin.predAbove 0 (e i)).predAbove (e x))
|
||||
rw [notInImage_oneEmbed]
|
||||
simp
|
||||
apply Subtype.ext
|
||||
simp
|
||||
change (e.subtypeEquivOfSubtype'
|
||||
((notInImage (oneEmbed i)).toFun ((Fin.predAbove 0 i).predAbove x))).1 = _
|
||||
rw [notInImage_oneEmbed]
|
||||
rw [Equiv.subtypeEquivOfSubtype', Equiv.subtypeEquivOfSubtype]
|
||||
simp
|
||||
rw [succAbove_predAbove_predAbove, succAbove_predAbove_predAbove]
|
||||
exact e.apply_eq_iff_eq.mp.mt (fun a => h a.symm)
|
||||
exact fun a => h a.symm
|
||||
exact e.apply_eq_iff_eq.mp.mt (fun a => h a.symm)
|
||||
simp only [Sum.map_inr, id_eq, Sum.inr.injEq, Equiv.symm_apply_eq,
|
||||
Function.Embedding.toEquivRange_apply, Function.Embedding.trans_apply, Equiv.coe_toEmbedding,
|
||||
Subtype.mk.injEq, EmbeddingLike.apply_eq_iff_eq]
|
||||
change x = f.toEquivRange _
|
||||
rw [Equiv.apply_symm_apply]
|
||||
· rw [markEmbeddingSet_on_not_mem f x hx,
|
||||
markEmbeddingSet_on_not_mem g (e x) (by simpa [← he] using hx)]
|
||||
subst he
|
||||
rfl
|
||||
|
||||
lemma markEmbedding_mapIso_left {n m : ℕ} (e : Fin n ≃ Fin m) (f : Fin n ↪ X) (g : Fin m ↪ X)
|
||||
(he : e.symm.toEmbedding.trans f = g) (T : RealLorentzTensor d X) :
|
||||
markEmbedding g T = mapIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by
|
||||
simpa [← he] using Equiv.forall_congr_left e)) e) (markEmbedding f T) := by
|
||||
rw [markEmbedding, markEmbedding]
|
||||
erw [← Equiv.trans_apply]
|
||||
rw [mapIso_trans]
|
||||
apply congrFun
|
||||
repeat apply congrArg
|
||||
refine Equiv.ext (fun x => ?_)
|
||||
simp only [Equiv.trans_apply, Equiv.sumCongr_apply]
|
||||
by_cases hx : x ∈ Finset.image f Finset.univ
|
||||
· rw [markEmbeddingSet_on_mem f x hx, markEmbeddingSet_on_mem g x (by
|
||||
simp [← he, hx]
|
||||
obtain ⟨y, _, hy2⟩ := Finset.mem_image.mp hx
|
||||
use e y
|
||||
simpa using hy2)]
|
||||
subst he
|
||||
simp [Equiv.symm_apply_eq]
|
||||
change x = f.toEquivRange _
|
||||
rw [Equiv.apply_symm_apply]
|
||||
· rw [markEmbeddingSet_on_not_mem f x hx, markEmbeddingSet_on_not_mem g x (by
|
||||
simpa [← he, hx] using fun x => not_exists.mp (Finset.mem_image.mpr.mt hx) (e.symm x))]
|
||||
subst he
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Marking a single element
|
||||
|
||||
-/
|
||||
|
||||
/-- An embedding from `Fin 1` into `X` given an element `x ∈ X`. -/
|
||||
@[simps!]
|
||||
def embedSingleton (x : X) : Fin 1 ↪ X :=
|
||||
⟨![x], fun x y => by fin_cases x; fin_cases y; simp⟩
|
||||
|
||||
lemma embedSingleton_toEquivRange_symm (x : X) :
|
||||
(embedSingleton x).toEquivRange.symm ⟨x, by simp⟩ = 0 := by
|
||||
exact Fin.fin_one_eq_zero _
|
||||
|
||||
/-- Equivalence, taking a tensor to a tensor with a single marked index. -/
|
||||
@[simps!]
|
||||
def markSingle (x : X) : RealLorentzTensor d X ≃ Marked d {x' // x' ≠ x} 1 :=
|
||||
(markEmbedding (embedSingleton x)).trans
|
||||
(mapIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by simp)) (Equiv.refl _)))
|
||||
|
||||
/-- Equivalence between index values of a tensor and the corrsponding tensor with a single
|
||||
marked index. -/
|
||||
@[simps!]
|
||||
def markSingleIndexValue (T : RealLorentzTensor d X) (x : X) :
|
||||
IndexValue d T.color ≃ IndexValue d (markSingle x T).color :=
|
||||
(markEmbeddingIndexValue (embedSingleton x) T).trans <|
|
||||
indexValueIso d (Equiv.sumCongr (Equiv.subtypeEquivRight (fun x => by simp)) (Equiv.refl _))
|
||||
(by funext x_1; simp)
|
||||
|
||||
/-- Given an equivalence of types, taking `x` to `y` the corresponding equivalence of
|
||||
subtypes of elements not equal to `x` and not equal to `y` respectively. -/
|
||||
@[simps!]
|
||||
def equivSingleCompl (e : X ≃ Y) {x : X} {y : Y} (he : e x = y) :
|
||||
{x' // x' ≠ x} ≃ {y' // y' ≠ y} :=
|
||||
(Equiv.subtypeEquivOfSubtype' e).trans <|
|
||||
Equiv.subtypeEquivRight (fun a => by simp [Equiv.symm_apply_eq, he])
|
||||
|
||||
lemma markSingle_mapIso (e : X ≃ Y) (x : X) (y : Y) (he : e x = y)
|
||||
(T : RealLorentzTensor d X) : markSingle y (mapIso d e T) =
|
||||
mapIso d (Equiv.sumCongr (equivSingleCompl e he) (Equiv.refl _)) (markSingle x T) := by
|
||||
rw [markSingle, Equiv.trans_apply]
|
||||
rw [markEmbedding_mapIso_right e (embedSingleton x) (embedSingleton y)
|
||||
(Function.Embedding.ext_iff.mp (fun a => by simpa using he)), markSingle, markEmbedding]
|
||||
erw [← Equiv.trans_apply, ← Equiv.trans_apply, ← Equiv.trans_apply]
|
||||
rw [mapIso_trans, mapIso_trans, mapIso_trans, mapIso_trans]
|
||||
apply congrFun
|
||||
repeat apply congrArg
|
||||
refine Equiv.ext fun x => ?_
|
||||
simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.sumCongr_trans,
|
||||
Equiv.trans_refl, Equiv.trans_apply, Equiv.sumCongr_apply, Equiv.coe_trans, Equiv.coe_refl,
|
||||
Sum.map_map, CompTriple.comp_eq]
|
||||
subst he
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Marking two elements
|
||||
|
||||
-/
|
||||
|
||||
/-- An embedding from `Fin 2` given two inequivalent elements. -/
|
||||
@[simps!]
|
||||
def embedDoubleton (x y : X) (h : x ≠ y) : Fin 2 ↪ X :=
|
||||
⟨![x, y], fun a b => by
|
||||
fin_cases a <;> fin_cases b <;> simp [h]
|
||||
exact h.symm⟩
|
||||
|
||||
end markingElements
|
||||
|
||||
end Marked
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue