Theorem: MvQPF.liftR_map (Qpf/Qpf/Multivariate/Constructions/Cofix.lean)

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ LiftR' R (f <$$> x) (g <$$> x)
Tactic:
rw [LiftR_def]
State after:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ ∃ u, (prod.fst ⊚ subtypeVal R) <$$> u = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> u = g <$$> x

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ ∃ u, (prod.fst ⊚ subtypeVal R) <$$> u = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> u = g <$$> x
Tactic:
exists h <$$> x
State after:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (prod.fst ⊚ subtypeVal R) <$$> h <$$> x = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> h <$$> x = g <$$> x

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (prod.fst ⊚ subtypeVal R) <$$> h <$$> x = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> h <$$> x = g <$$> x
Tactic:
rw [MvFunctor.map_map, comp_assoc, hh, ← comp_assoc, fst_prod_mk, comp_assoc, fst_diag]
State after:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> h <$$> x = g <$$> x

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> h <$$> x = g <$$> x
Tactic:
rw [MvFunctor.map_map, comp_assoc, hh, ← comp_assoc, snd_prod_mk, comp_assoc, snd_diag]
State after:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (g ⊚ TypeVec.id) <$$> x = g <$$> x

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (g ⊚ TypeVec.id) <$$> x = g <$$> x
Tactic:
dsimp [LiftR']
State after:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ f <$$> x = f <$$> x ∧ g <$$> x = g <$$> x

State before:
n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ repeat n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ f <$$> x = f <$$> x ∧ g <$$> x = g <$$> x
Tactic:
constructor <;> rfl
State after:
no goals