Module: Mathlib.Data.PFunctor.Multivariate.M

Goal state before:
pp: case refine_2.rel
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ y✝ : P.M α
a✝ : R x✝ y✝
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝

pp: case refine_2.refl
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ : P.M α
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝

pp: case refine_2.symm
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ y✝ : P.M α
a✝ : Relation.EqvGen R x✝ y✝
a_ih✝ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝

pp: case refine_2.trans
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ y✝ z✝ : P.M α
a✝¹ : Relation.EqvGen R x✝ y✝
a✝ : Relation.EqvGen R y✝ z✝
a_ih✝¹ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝
a_ih✝ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P z✝
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P z✝

Tactic:
· rw [← Quot.factor_mk_eq R (Relation.EqvGen R) this]
      rwa [appendFun_comp_id, ← MvFunctor.map_map, ← MvFunctor.map_map, h]

Goal state after:
case refine_2.refl
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ : P.M α
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝

case refine_2.symm
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ y✝ : P.M α
a✝ : Relation.EqvGen R x✝ y✝
a_ih✝ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝

case refine_2.trans
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
R : P.M α → P.M α → Prop
h : ∀ (x y : P.M α), R x y → (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y
x y : P.M α
this : ∀ (x y : P.M α), R x y → Relation.EqvGen R x y
x✝ y✝ z✝ : P.M α
a✝¹ : Relation.EqvGen R x✝ y✝
a✝ : Relation.EqvGen R y✝ z✝
a_ih✝¹ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝
a_ih✝ :
  (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P y✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P z✝
⊢ (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P x✝ =
    (TypeVec.id ::: Quot.mk (Relation.EqvGen R)) <$$> dest P z✝