Theorem: Dioph.diophFn_compn
File: Mathlib/NumberTheory/Dioph.lean

tactic: dsimp
state_before: α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
⊢ v ∈ {v | v ∘ (id ⊗ elim0) ∈ S} ↔ v ∈ {v | (v ⊗ fun i => f i v) ∈ S}
state_after: α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
⊢ v ∘ (id ⊗ elim0) ∈ S ↔ (v ⊗ fun i => f i v) ∈ S

tactic: refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
state_before: α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
⊢ v ∘ (id ⊗ elim0) ∈ S ↔ (v ⊗ fun i => f i v) ∈ S
state_after: α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
⊢ v ∘ (id ⊗ elim0) = v ⊗ fun i => f i v

tactic: ext x
state_before: α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
⊢ v ∘ (id ⊗ elim0) = v ⊗ fun i => f i v
state_after: case h
α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
x : α ⊕ Fin2 0
⊢ (v ∘ (id ⊗ elim0)) x = (v ⊗ fun i => f i v) x

tactic: obtain _ | _ | _ := x
state_before: case h
α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
x : α ⊕ Fin2 0
⊢ (v ∘ (id ⊗ elim0)) x = (v ⊗ fun i => f i v) x
state_after: case h.inl
α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
val✝ : α
⊢ (v ∘ (id ⊗ elim0)) (inl val✝) = (v ⊗ fun i => f i v) (inl val✝)

tactic: rfl
state_before: case h.inl
α β : Type
n : ℕ
S : Set (α ⊕ Fin2 0 → ℕ)
d : Dioph S
f : Vector3 ((α → ℕ) → ℕ) 0
x✝ : VectorAllP DiophFn f
v : α → ℕ
val✝ : α
⊢ (v ∘ (id ⊗ elim0)) (inl val✝) = (v ⊗ fun i => f i v) (inl val✝)
state_after: no goals

tactic: simp only [vectorAllP_cons, and_imp]
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
⊢ VectorAllP DiophFn (f :: fl) → Dioph {v | (v ⊗ fun i => (f :: fl) i v) ∈ S}
state_after: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
⊢ DiophFn f → VectorAllP DiophFn fl → Dioph {v | (v ⊗ fun i => (f :: fl) i v) ∈ S}

tactic: exact fun df dfl =>
  have : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr) ∈ S} :=
    ext (diophFn_comp1 (reindex_dioph _ (some ∘ inl ⊗ none::some ∘ inr) d) <|
        reindex_diophFn inl df)
      fun v => by
        dsimp
        refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
        ext x; obtain _ | _ | _ := x <;> rfl
  have : Dioph {v | (v ⊗ f v::fun i : Fin2 n => fl i v) ∈ S} :=
    @diophFn_compn n (fun v => S (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr)) this _ dfl
  ext this fun v => by
    dsimp;
    refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
    ext x; obtain _ | _ | _ := x <;> rfl
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
⊢ DiophFn f → VectorAllP DiophFn fl → Dioph {v | (v ⊗ fun i => (f :: fl) i v) ∈ S}
state_after: no goals

tactic: dsimp
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
⊢ v ∈ {v | f (v ∘ inl) ::ₒ v ∈ {v | v ∘ (some ∘ inl ⊗ none :: some ∘ inr) ∈ S}} ↔
    v ∈ {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
state_after: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
⊢ (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) ∈ S ↔ (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S

tactic: refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
⊢ (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) ∈ S ↔ (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S
state_after: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
⊢ (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) = v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr

tactic: ext x
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
⊢ (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) = v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr
state_after: case h
α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
x : α ⊕ Fin2 (n + 1)
⊢ ((f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr)) x = (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) x

tactic: obtain _ | _ | _ := x <;> rfl
state_before: case h
α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
v : α ⊕ Fin2 n → ℕ
x : α ⊕ Fin2 (n + 1)
⊢ ((f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr)) x = (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) x
state_after: no goals

tactic: dsimp
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
⊢ v ∈ {v | (v ⊗ f v :: fun i => fl i v) ∈ S} ↔ v ∈ {v | (v ⊗ fun i => (f :: fl) i v) ∈ S}
state_after: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
⊢ (v ⊗ f v :: fun i => fl i v) ∈ S ↔ (v ⊗ fun i => (f :: fl) i v) ∈ S

tactic: refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
⊢ (v ⊗ f v :: fun i => fl i v) ∈ S ↔ (v ⊗ fun i => (f :: fl) i v) ∈ S
state_after: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
⊢ (v ⊗ f v :: fun i => fl i v) = v ⊗ fun i => (f :: fl) i v

tactic: ext x
state_before: α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
⊢ (v ⊗ f v :: fun i => fl i v) = v ⊗ fun i => (f :: fl) i v
state_after: case h
α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
x : α ⊕ Fin2 (n + 1)
⊢ (v ⊗ f v :: fun i => fl i v) x = (v ⊗ fun i => (f :: fl) i v) x

tactic: obtain _ | _ | _ := x <;> rfl
state_before: case h
α β : Type
n✝ n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
df : DiophFn f
dfl : VectorAllP DiophFn fl
this✝ : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr) ∈ S}
this : Dioph {v | (v ⊗ f v :: fun i => fl i v) ∈ S}
v : α → ℕ
x : α ⊕ Fin2 (n + 1)
⊢ (v ⊗ f v :: fun i => fl i v) x = (v ⊗ fun i => (f :: fl) i v) x
state_after: no goals