[GOAL]
α : Type u_1
inst✝ : Primcodable α
p : α → Prop
⊢ ∀ (a : α), p a ↔ p (id a)
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : p a
⊢ r ((g ∘ f) a)
[PROOFSTEP]
erw [← h₂, ← h₁]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : p a
⊢ p a
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : r ((g ∘ f) a)
⊢ p a
[PROOFSTEP]
rwa [h₁, h₂]
[GOAL]
α : Type u_1
inst✝ : Primcodable α
p : α → Prop
⊢ ∀ (a : α), p a ↔ p (id a)
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
i₁ : Injective f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
i₂ : Injective g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : p a
⊢ r ((g ∘ f) a)
[PROOFSTEP]
erw [← h₂, ← h₁]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
i₁ : Injective f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
i₂ : Injective g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : p a
⊢ p a
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → β
c₁ : Computable f
i₁ : Injective f
h₁ : ∀ (a : α), p a ↔ q (f a)
g : β → γ
c₂ : Computable g
i₂ : Injective g
h₂ : ∀ (a : β), q a ↔ r (g a)
a : α
h : r ((g ∘ f) a)
⊢ p a
[PROOFSTEP]
rwa [h₁, h₂]
[GOAL]
α : Type u_1
β : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable β
e : α ≃ β
q : β → Prop
h : Computable ↑e.symm
⊢ q ≤₁ (q ∘ ↑e)
[PROOFSTEP]
convert OneOneReducible.of_equiv _ h
[GOAL]
case h.e'_5
α : Type u_1
β : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable β
e : α ≃ β
q : β → Prop
h : Computable ↑e.symm
⊢ q = (q ∘ ↑e) ∘ ↑e.symm
[PROOFSTEP]
funext
[GOAL]
case h.e'_5.h
α : Type u_1
β : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable β
e : α ≃ β
q : β → Prop
h : Computable ↑e.symm
x✝ : β
⊢ q x✝ = ((q ∘ ↑e) ∘ ↑e.symm) x✝
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
q : β → Prop
h₁ : p ≤₀ q
h₂ : ComputablePred q
⊢ ComputablePred p
[PROOFSTEP]
rcases h₁ with ⟨f, c, hf⟩
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
q : β → Prop
h₂ : ComputablePred q
f : α → β
c : Computable f
hf : ∀ (a : α), p a ↔ q (f a)
⊢ ComputablePred p
[PROOFSTEP]
rw [show p = fun a => q (f a) from Set.ext hf]
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
q : β → Prop
h₂ : ComputablePred q
f : α → β
c : Computable f
hf : ∀ (a : α), p a ↔ q (f a)
⊢ ComputablePred fun a => q (f a)
[PROOFSTEP]
rcases computable_iff.1 h₂ with ⟨g, hg, rfl⟩
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
f : α → β
c : Computable f
g : β → Bool
hg : Computable g
h₂ : ComputablePred fun a => g a = true
hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)
⊢ ComputablePred fun a => (fun a => g a = true) (f a)
[PROOFSTEP]
exact ⟨by infer_instance, by simpa using hg.comp c⟩
[GOAL]
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
f : α → β
c : Computable f
g : β → Bool
hg : Computable g
h₂ : ComputablePred fun a => g a = true
hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)
⊢ DecidablePred fun a => (fun a => g a = true) (f a)
[PROOFSTEP]
infer_instance
[GOAL]
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
f : α → β
c : Computable f
g : β → Bool
hg : Computable g
h₂ : ComputablePred fun a => g a = true
hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)
⊢ Computable fun a => decide ((fun a => (fun a => g a = true) (f a)) a)
[PROOFSTEP]
simpa using hg.comp c
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → γ
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ r (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
x : α ⊕ β
⊢ (p ⊕' q) x ↔ r ((f ⊕' g) x)
[PROOFSTEP]
cases x <;> [apply h₁; apply h₂]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → γ
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ r (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
x : α ⊕ β
⊢ (p ⊕' q) x ↔ r ((f ⊕' g) x)
[PROOFSTEP]
cases x
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → γ
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ r (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
val✝ : α
⊢ (p ⊕' q) (Sum.inl val✝) ↔ r ((f ⊕' g) (Sum.inl val✝))
[PROOFSTEP]
apply h₁
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
p : α → Prop
q : β → Prop
r : γ → Prop
f : α → γ
c₁ : Computable f
h₁ : ∀ (a : α), p a ↔ r (f a)
g : β → γ
c₂ : Computable g
h₂ : ∀ (a : β), q a ↔ r (g a)
val✝ : β
⊢ (p ⊕' q) (Sum.inr val✝) ↔ r ((f ⊕' g) (Sum.inr val✝))
[PROOFSTEP]
apply h₂
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : Set α
⊢ ∀ (a : α), p a ↔ toNat p (Encodable.encode a)
[PROOFSTEP]
simp [toNat, setOf]
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : Set α
⊢ ManyOneEquiv (toNat p) p
[PROOFSTEP]
simp [ManyOneEquiv]
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : Set α
q : Set β
⊢ ManyOneEquiv (toNat p) (toNat q) ↔ ManyOneEquiv p q
[PROOFSTEP]
simp [ManyOneEquiv]
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ d₂ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p : Set ℕ
q₁ q₂ : ℕ → Prop
hq : ManyOneEquiv q₁ q₂
⊢ ManyOneEquiv p p
[PROOFSTEP]
rfl
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ d₂ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
⊢ ∀ (p q : ℕ → Prop),
    ManyOneEquiv p q →
      (fun p => ManyOneDegree.liftOn d₂ (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) p =
        (fun p => ManyOneDegree.liftOn d₂ (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) q
[PROOFSTEP]
intro p₁ p₂ hp
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ d₂ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p₁ p₂ : ℕ → Prop
hp : ManyOneEquiv p₁ p₂
⊢ (fun p => ManyOneDegree.liftOn d₂ (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) p₁ =
    (fun p => ManyOneDegree.liftOn d₂ (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) p₂
[PROOFSTEP]
induction d₂ using ManyOneDegree.ind_on
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p₁ p₂ : ℕ → Prop
hp : ManyOneEquiv p₁ p₂
p✝ : Set ℕ
⊢ (fun p => ManyOneDegree.liftOn (of p✝) (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) p₁ =
    (fun p => ManyOneDegree.liftOn (of p✝) (f p) (_ : ∀ (q₁ q₂ : ℕ → Prop), ManyOneEquiv q₁ q₂ → f p q₁ = f p q₂)) p₂
[PROOFSTEP]
apply h
[GOAL]
case h.a
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p₁ p₂ : ℕ → Prop
hp : ManyOneEquiv p₁ p₂
p✝ : Set ℕ
⊢ ManyOneEquiv p₁ p₂
case h.a
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p₁ p₂ : ℕ → Prop
hp : ManyOneEquiv p₁ p₂
p✝ : Set ℕ
⊢ ManyOneEquiv (toNat p✝) (toNat p✝)
[PROOFSTEP]
assumption
[GOAL]
case h.a
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
φ : Sort ?u.72115
d₁ : ManyOneDegree
f : Set ℕ → Set ℕ → φ
h : ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop), ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → f p₁ q₁ = f p₂ q₂
p₁ p₂ : ℕ → Prop
hp : ManyOneEquiv p₁ p₂
p✝ : Set ℕ
⊢ ManyOneEquiv (toNat p✝) (toNat p✝)
[PROOFSTEP]
rfl
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : α → Prop
q : β → Prop
⊢ of p = of q ↔ ManyOneEquiv p q
[PROOFSTEP]
rw [of, of, Quotient.eq'']
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : α → Prop
q : β → Prop
⊢ Setoid.r (toNat p) (toNat q) ↔ ManyOneEquiv p q
[PROOFSTEP]
unfold Setoid.r
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p : α → Prop
q : β → Prop
⊢ { r := ManyOneEquiv, iseqv := proof_1 }.1 (toNat p) (toNat q) ↔ ManyOneEquiv p q
[PROOFSTEP]
simp
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d : ManyOneDegree
⊢ d ≤ d
[PROOFSTEP]
induction d using ManyOneDegree.ind_on
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝ : Set ℕ
⊢ of p✝ ≤ of p✝
[PROOFSTEP]
simp
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝ : Set ℕ
⊢ p✝ ≤₀ p✝
[PROOFSTEP]
rfl
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
⊢ d₁ ≤ d₂ → d₂ ≤ d₁ → d₁ = d₂
[PROOFSTEP]
induction d₁ using ManyOneDegree.ind_on
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₂ : ManyOneDegree
p✝ : Set ℕ
⊢ of p✝ ≤ d₂ → d₂ ≤ of p✝ → of p✝ = d₂
[PROOFSTEP]
induction d₂ using ManyOneDegree.ind_on
[GOAL]
case h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝¹ p✝ : Set ℕ
⊢ of p✝¹ ≤ of p✝ → of p✝ ≤ of p✝¹ → of p✝¹ = of p✝
[PROOFSTEP]
intro hp hq
[GOAL]
case h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝¹ p✝ : Set ℕ
hp : of p✝¹ ≤ of p✝
hq : of p✝ ≤ of p✝¹
⊢ of p✝¹ = of p✝
[PROOFSTEP]
simp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and_iff]
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ d₃ : ManyOneDegree
⊢ d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃
[PROOFSTEP]
induction d₁ using ManyOneDegree.ind_on
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₂ d₃ : ManyOneDegree
p✝ : Set ℕ
⊢ of p✝ ≤ d₂ → d₂ ≤ d₃ → of p✝ ≤ d₃
[PROOFSTEP]
induction d₂ using ManyOneDegree.ind_on
[GOAL]
case h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₃ : ManyOneDegree
p✝¹ p✝ : Set ℕ
⊢ of p✝¹ ≤ of p✝ → of p✝ ≤ d₃ → of p✝¹ ≤ d₃
[PROOFSTEP]
induction d₃ using ManyOneDegree.ind_on
[GOAL]
case h.h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝² p✝¹ p✝ : Set ℕ
⊢ of p✝² ≤ of p✝¹ → of p✝¹ ≤ of p✝ → of p✝² ≤ of p✝
[PROOFSTEP]
apply ManyOneReducible.trans
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
⊢ ∀ (p₁ p₂ q₁ q₂ : ℕ → Prop),
    ManyOneEquiv p₁ p₂ → ManyOneEquiv q₁ q₂ → (fun a b => of (a ⊕' b)) p₁ q₁ = (fun a b => of (a ⊕' b)) p₂ q₂
[PROOFSTEP]
rintro a b c d ⟨hl₁, hr₁⟩ ⟨hl₂, hr₂⟩
[GOAL]
case intro.intro
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
a b c d : ℕ → Prop
hl₁ : a ≤₀ b
hr₁ : b ≤₀ a
hl₂ : c ≤₀ d
hr₂ : d ≤₀ c
⊢ (fun a b => of (a ⊕' b)) a c = (fun a b => of (a ⊕' b)) b d
[PROOFSTEP]
rw [of_eq_of]
[GOAL]
case intro.intro
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
a b c d : ℕ → Prop
hl₁ : a ≤₀ b
hr₁ : b ≤₀ a
hl₂ : c ≤₀ d
hr₂ : d ≤₀ c
⊢ ManyOneEquiv (a ⊕' c) (b ⊕' d)
[PROOFSTEP]
exact
  ⟨disjoin_manyOneReducible (hl₁.trans OneOneReducible.disjoin_left.to_many_one)
      (hl₂.trans OneOneReducible.disjoin_right.to_many_one),
    disjoin_manyOneReducible (hr₁.trans OneOneReducible.disjoin_left.to_many_one)
      (hr₂.trans OneOneReducible.disjoin_right.to_many_one)⟩
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ d₃ : ManyOneDegree
⊢ d₁ + d₂ ≤ d₃ ↔ d₁ ≤ d₃ ∧ d₂ ≤ d₃
[PROOFSTEP]
induction d₁ using ManyOneDegree.ind_on
[GOAL]
case h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₂ d₃ : ManyOneDegree
p✝ : Set ℕ
⊢ of p✝ + d₂ ≤ d₃ ↔ of p✝ ≤ d₃ ∧ d₂ ≤ d₃
[PROOFSTEP]
induction d₂ using ManyOneDegree.ind_on
[GOAL]
case h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₃ : ManyOneDegree
p✝¹ p✝ : Set ℕ
⊢ of p✝¹ + of p✝ ≤ d₃ ↔ of p✝¹ ≤ d₃ ∧ of p✝ ≤ d₃
[PROOFSTEP]
induction d₃ using ManyOneDegree.ind_on
[GOAL]
case h.h.h
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
p✝² p✝¹ p✝ : Set ℕ
⊢ of p✝² + of p✝¹ ≤ of p✝ ↔ of p✝² ≤ of p✝ ∧ of p✝¹ ≤ of p✝
[PROOFSTEP]
simpa only [← add_of, of_le_of] using disjoin_le
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
⊢ d₁ + ?m.117648 d₁ d₂ ≤ d₁ + d₂
[PROOFSTEP]
rfl
[GOAL]
α : Type u
inst✝⁵ : Primcodable α
inst✝⁴ : Inhabited α
β : Type v
inst✝³ : Primcodable β
inst✝² : Inhabited β
γ : Type w
inst✝¹ : Primcodable γ
inst✝ : Inhabited γ
d₁ d₂ : ManyOneDegree
⊢ ?m.117829 d₁ d₂ + d₂ ≤ d₁ + d₂
[PROOFSTEP]
rfl
