lean-stuff/MutualNested.lean at main · cppio/lean-stuff · GitHub

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defMonotone (f : Prop → Prop) : Prop :=
 ∀ ⦃p p'⦄, (p → p') → f p → f p'

defMonotone₂ (f : Prop → Prop → Prop) : Prop :=
 ∀ ⦃p p' q q'⦄, (p → p') → (q → q') → f p q → f p' q'

namespace Inductive

-- ∀ p, (f p → p) → p
def μ f (hf : Monotone f) : Prop :=
 f (μ f hf)
inductive_fixpoint monotonicity hf

namespace μ

defrec : ∀ p, (f p → p) → μ f hf → p :=
 induct f hf

deffold : f (μ f hf) → μ f hf :=
 (eq_def f hf).mpr

defmono₁ (hf : Monotone₂ f) : Monotone fun p => μ (f p) fun _ _ => hf id :=
fun _ _ hp => rec _ <| fold ∘ hf hp id

defmono₂ (hf : Monotone₂ f) : Monotone fun q => μ (f · q) fun _ _ hp => hf hp id :=
fun _ _ hq => rec _ <| fold ∘ hf id hq

end μ

mutual

-- ∀ p q, (f p q → p) → (g p q → q) → p
defPfg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 f (P f g hf hg) (Q f g hf hg)
inductive_fixpoint monotonicity fun _ _ ⟨hp, hq⟩ => hf hp hq

-- ∀ p q, (f p q → p) → (g p q → q) → q
defQfg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 g (P f g hf hg) (Q f g hf hg)
inductive_fixpoint monotonicity fun _ _ ⟨hp, hq⟩ => hg hp hq

end

namespace P

unseal P in
defrecpq (hp : f p q → p) (hq : g p q → q) : P f g hf hg → p :=
fun ⟨_, ⟨_, h⟩, hp'⟩ => (h ⟨p, q⟩ ⟨hp, hq⟩).left hp'

deffold : f (P f g hf hg) (Q f g hf hg) → P f g hf hg :=
 (eq_def f g hf hg).mpr

end P

namespace Q

unseal Q in
defrecpq (hp : f p q → p) (hq : g p q → q) : Q f g hf hg → q :=
fun ⟨_, ⟨_, h⟩, hq'⟩ => (h ⟨p, q⟩ ⟨hp, hq⟩).right hq'

deffold : g (P f g hf hg) (Q f g hf hg) → Q f g hf hg :=
 (eq_def f g hf hg).mpr

end Q

defP₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun p => f p (μ (fun q => g p q) fun _ _ => hg id)) fun _ _ h => hf h (μ.mono₁ hg h)

defQ₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun q => g (P₁ f g hf hg) q) fun _ _ => hg id

defQ₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun q => g (μ (fun p => f p q) fun _ _ hp => hf hp id) q) fun _ _ h => hg (μ.mono₂ hf h) h

defP₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun p => f p (Q₂ f g hf hg)) fun _ _ hp => hf hp id

theoremnested₁_eq_nested₂ : (P₁ f g hf hg ↔ P₂ f g hf hg) ∧ (Q₁ f g hf hg ↔ Q₂ f g hf hg) where
 left.mp := μ.rec (P₂ ..) (μ.fold ∘ hf id (μ.rec (Q₂ ..) (μ.fold ∘ id)))
 left.mpr := μ.rec (P₁ ..) (μ.fold ∘ hf id (μ.rec (Q₁ ..) (μ.fold ∘ hg (μ.rec (P₁ f g hf hg) (μ.fold ∘ id)) id)))
 right.mp := μ.rec (Q₂ ..) (μ.fold ∘ hg (μ.rec (P₂ ..) (μ.fold ∘ hf id (μ.rec (Q₂ f g hf hg) (μ.fold ∘ id)))) id)
 right.mpr := μ.rec (Q₁ ..) (μ.fold ∘ hg (μ.rec (P₁ ..) (μ.fold ∘ id)) id)

theoremmutual_eq_nested₁ : (P f g hf hg ↔ P₁ f g hf hg) ∧ (Q f g hf hg ↔ Q₁ f g hf hg) where
 left.mp := P.rec (P₁ ..) (Q₁ ..) (μ.fold ∘ id) μ.fold
 left.mpr := μ.rec (P ..) (P.fold ∘ hf id (μ.rec (Q ..) Q.fold))
 right.mp := Q.rec (P₁ ..) (Q₁ ..) (μ.fold ∘ id) μ.fold
 right.mpr := μ.rec (Q ..) (Q.fold ∘ hg (μ.rec (P ..) (P.fold ∘ hf id (μ.rec (Q ..) Q.fold))) id)

theoremmutual_eq_nested₂ : (P f g hf hg ↔ P₂ f g hf hg) ∧ (Q f g hf hg ↔ Q₂ f g hf hg) where
 left.mp := P.rec (P₂ ..) (Q₂ ..) (μ.fold ∘ id) (μ.fold ∘ id)
 left.mpr := μ.rec (P ..) (P.fold ∘ hf id (μ.rec (Q ..) (Q.fold ∘ hg (μ.rec (P ..) P.fold) id)))
 right.mp := Q.rec (P₂ ..) (Q₂ ..) (μ.fold ∘ id) (μ.fold ∘ id)
 right.mpr := μ.rec (Q ..) (Q.fold ∘ hg (μ.rec (P ..) P.fold) id)

defRf (hf : Monotone₂ f) : Prop :=
 μ (fun p => f p p) fun _ _ hp => hf hp hp

defR₁f (hf : Monotone₂ f) : Prop :=
 μ (fun p => μ (fun q => f p q) fun _ _ => hf id) (μ.mono₁ hf)

defR₂f (hf : Monotone₂ f) : Prop :=
 μ (fun q => μ (fun p => f p q) fun _ _ hp => hf hp id) (μ.mono₂ hf)

theoremnested₁_eq_nested₂' : R₁ f hf ↔ R₂ f hf where
 mp := μ.fold ∘ μ.rec _ (μ.rec _ (μ.fold ∘ hf id (μ.fold ∘ id)))
 mpr := μ.fold ∘ μ.rec _ (μ.rec _ (μ.fold ∘ hf (μ.fold ∘ id) id))

theoremflat_eq_nested₁ : R f hf ↔ R₁ f hf where
 mp := μ.fold ∘ μ.rec _ (μ.fold ∘ hf (μ.fold ∘ id) id)
 mpr := μ.rec (R ..) (μ.rec (R ..) (μ.fold ∘ id))

theoremflat_eq_nested₂ : R f hf ↔ R₂ f hf where
 mp := μ.fold ∘ μ.rec _ (μ.fold ∘ hf id (μ.fold ∘ id))
 mpr := μ.rec (R ..) (μ.rec (R ..) (μ.fold ∘ id))

end Inductive

namespace Coinductive

-- ∃ p, (p → f p) ∧ p
def ν f (hf : Monotone f) : Prop :=
 f (ν f hf)
coinductive_fixpoint monotonicity fun _ _ h => hf h

namespace ν

defcorec : ∀ p, (p → f p) → p → ν f hf :=
 coinduct f hf

defunfold : ν f hf → f (ν f hf) :=
 (eq_def f hf).mp

defmono₁ (hf : Monotone₂ f) : Monotone fun p => ν (f p) fun _ _ => hf id :=
fun _ _ hp => corec _ <| hf hp id ∘ unfold

defmono₂ (hf : Monotone₂ f) : Monotone fun q => ν (f · q) fun _ _ hp => hf hp id :=
fun _ _ hq => corec _ <| hf id hq ∘ unfold

end ν

mutual

-- ∃ p q, (p → f p q) ∧ (q → g p q) ∧ p
defPfg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 f (P f g hf hg) (Q f g hf hg)
coinductive_fixpoint monotonicity fun _ _ ⟨hp, hq⟩ => hf hp hq

-- ∃ p q, (p → f p q) ∧ (q → g p q) ∧ q
defQfg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 g (P f g hf hg) (Q f g hf hg)
coinductive_fixpoint monotonicity fun _ _ ⟨hp, hq⟩ => hg hp hq

end

namespace P

unseal P in
defcorecpq (hp : p → f p q) (hq : q → g p q) (hp' : p) : P f g hf hg :=
fun _ ⟨_, h⟩ => (h ⟨p, q⟩ ⟨hp, hq⟩).left hp'

defunfold : P f g hf hg → f (P f g hf hg) (Q f g hf hg) :=
 (eq_def f g hf hg).mp

end P

namespace Q

unseal Q in
defcorecpq (hp : p → f p q) (hq : q → g p q) (hq' : q) : Q f g hf hg :=
fun _ ⟨_, h⟩ => (h ⟨p, q⟩ ⟨hp, hq⟩).right hq'

defunfold : Q f g hf hg → g (P f g hf hg) (Q f g hf hg) :=
 (eq_def f g hf hg).mp

end Q

defP₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun p => f p (ν (fun q => g p q) fun _ _ => hg id)) fun _ _ h => hf h (ν.mono₁ hg h)

defQ₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun q => g (P₁ f g hf hg) q) fun _ _ => hg id

defQ₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun q => g (ν (fun p => f p q) fun _ _ hp => hf hp id) q) fun _ _ h => hg (ν.mono₂ hf h) h

defP₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun p => f p (Q₂ f g hf hg)) fun _ _ hp => hf hp id

theoremnested₁_eq_nested₂ : (P₁ f g hf hg ↔ P₂ f g hf hg) ∧ (Q₁ f g hf hg ↔ Q₂ f g hf hg) where
 left.mp := ν.corec (P₁ ..) (hf id (ν.corec (Q₁ ..) (hg (ν.corec (P₁ f g hf hg) ν.unfold) id ∘ ν.unfold)) ∘ ν.unfold)
 left.mpr := ν.corec (P₂ ..) (hf id (ν.corec (Q₂ ..) ν.unfold) ∘ ν.unfold)
 right.mp := ν.corec (Q₁ ..) (hg (ν.corec (P₁ ..) ν.unfold) id ∘ ν.unfold)
 right.mpr := ν.corec (Q₂ ..) (hg (ν.corec (P₂ ..) (hf id (ν.corec (Q₂ f g hf hg) ν.unfold) ∘ ν.unfold)) id ∘ ν.unfold)

theoremmutual_eq_nested₁ : (P f g hf hg ↔ P₁ f g hf hg) ∧ (Q f g hf hg ↔ Q₁ f g hf hg) where
 left.mp := ν.corec (P ..) (hf id (ν.corec (Q ..) Q.unfold) ∘ P.unfold)
 left.mpr := P.corec (P₁ ..) (Q₁ ..) ν.unfold ν.unfold
 right.mp := ν.corec (Q ..) (hg (ν.corec (P ..) (hf id (ν.corec (Q ..) Q.unfold) ∘ P.unfold)) id ∘ Q.unfold)
 right.mpr := Q.corec (P₁ ..) (Q₁ ..) ν.unfold ν.unfold

theoremmutual_eq_nested₂ : (P f g hf hg ↔ P₂ f g hf hg) ∧ (Q f g hf hg ↔ Q₂ f g hf hg) where
 left.mp := ν.corec (P ..) (hf id (ν.corec (Q ..) (hg (ν.corec (P ..) P.unfold) id ∘ Q.unfold)) ∘ P.unfold)
 left.mpr := P.corec (P₂ ..) (Q₂ ..) ν.unfold ν.unfold
 right.mp := ν.corec (Q ..) (hg (ν.corec (P ..) P.unfold) id ∘ Q.unfold)
 right.mpr := Q.corec (P₂ ..) (Q₂ ..) ν.unfold ν.unfold

defRf (hf : Monotone₂ f) : Prop :=
 ν (fun p => f p p) fun _ _ hp => hf hp hp

defR₁f (hf : Monotone₂ f) : Prop :=
 ν (fun p => ν (fun q => f p q) fun _ _ => hf id) (ν.mono₁ hf)

defR₂f (hf : Monotone₂ f) : Prop :=
 ν (fun q => ν (fun p => f p q) fun _ _ hp => hf hp id) (ν.mono₂ hf)

theoremnested₁_eq_nested₂' : R₁ f hf ↔ R₂ f hf where
 mp := flip (· ∘ ·) ν.unfold <| ν.corec _ (ν.corec _ (hf ν.unfold id ∘ ν.unfold))
 mpr := flip (· ∘ ·) ν.unfold <| ν.corec _ (ν.corec _ (hf id ν.unfold ∘ ν.unfold))

theoremflat_eq_nested₁ : R f hf ↔ R₁ f hf where
 mp := ν.corec (R ..) (ν.corec (R ..) ν.unfold)
 mpr := flip (· ∘ ·) ν.unfold <| ν.corec _ (hf ν.unfold id ∘ ν.unfold)

theoremflat_eq_nested₂ : R f hf ↔ R₂ f hf where
 mp := ν.corec (R ..) (ν.corec (R ..) ν.unfold)
 mpr := flip (· ∘ ·) ν.unfold <| ν.corec _ (hf id ν.unfold ∘ ν.unfold)

end Coinductive

namespace Mixed

open Inductive (μ)
open Coinductive (ν)

defP₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun p => f p (ν (fun q => g p q) fun _ _ => hg id)) fun _ _ h => hf h (ν.mono₁ hg h)

defQ₁fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun q => g (P₁ f g hf hg) q) fun _ _ => hg id

defQ₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 ν (fun q => g (μ (fun p => f p q) fun _ _ hp => hf hp id) q) fun _ _ h => hg (μ.mono₂ hf h) h

defP₂fg (hf : Monotone₂ f) (hg : Monotone₂ g) : Prop :=
 μ (fun p => f p (Q₂ f g hf hg)) fun _ _ hp => hf hp id

theoremnested₁_ne_nested₂ : ∃ f g hf hg, ¬P₁ f g hf hg ∧ P₂ f g hf hg :=
 ⟨fun _ q => q, fun p _ => p, fun _ _ _ _ _ => id, fun _ _ _ _ hp _ => hp, μ.rec _ ν.unfold, μ.fold (ν.corec True (μ.fold ∘ id) ⟨⟩)⟩

theoremnested₁_ne_nested₂' : ∃ f g hf hg, ¬Q₁ f g hf hg ∧ Q₂ f g hf hg :=
 ⟨fun _ q => q, fun p _ => p, fun _ _ _ _ _ => id, fun _ _ _ _ hp _ => hp, μ.rec _ ν.unfold ∘ ν.unfold, ν.corec True (μ.fold ∘ id) ⟨⟩⟩

defR₁f (hf : Monotone₂ f) : Prop :=
 μ (fun p => ν (fun q => f p q) fun _ _ => hf id) (ν.mono₁ hf)

defR₂f (hf : Monotone₂ f) : Prop :=
 ν (fun q => μ (fun p => f p q) fun _ _ hp => hf hp id) (μ.mono₂ hf)

example : R₁ f hf → R₂ f hf :=
 ν.corec _ (μ.rec _ (μ.fold ∘ hf id (μ.fold ∘ ν.corec _ (hf (μ.rec (R₁ f hf) (by exact μ.fold ∘ ν.corec _ (hf id (hf id μ.fold ∘ ν.unfold ∘ μ.rec (ν (fun q => f (R₁ f hf) q) _) (ν.mono₁ hf (μ.fold ∘ id)))))) id ∘ ν.unfold)) ∘ ν.unfold))

end Mixed