internlm/Lean-Github huggingface dataset:
Theorem: IQPF.M.bisim_lemma
Proof state before:
case a.a.mk.mk
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝ : I
x✝ y✝ : IContainer.M (C F) i✝
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i : I
x y : IContainer.M (C F) i
h₂✝ : r' i x y
nx : IContainer.A (C F) i
kx : (y : IContainer.B (C F) i nx) → IContainer.M (C F) (IContainer.N (C F) i nx y)
ny : IContainer.A (C F) i
ky : (y : IContainer.B (C F) i ny) → IContainer.M (C F) (IContainer.N (C F) i ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
f : (i : I) → Quot (r i) → Quot (r' i) :=
  fun i =>
    Quot.lift
      (Quot.lift (Quot.mk (r' i))
        (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b))
      (_ :
        ∀ (x b : Quot (congr F i)),
          r i x b →
            Quot.lift (Quot.mk (r' i))
                (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) x =
              Quot.lift (Quot.mk (r' i))
                (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b)
this : ∀ (i : I), f i ∘ Quot.mk (r i) ∘ Quot.mk (congr F i) = Quot.mk (r' i)
⊢ abs
      (IContainer.Map (fun i => f i)
        (IContainer.Map (fun i => Quot.mk (r i))
          (IContainer.Map (fun i => Quot.mk (congr F i)) { fst := nx, snd := kx }))) =
    abs
      (IContainer.Map (fun i => f i)
        (IContainer.Map (fun i => Quot.mk (r i))
          (IContainer.Map (fun i => Quot.mk (congr F i)) { fst := ny, snd := ky })))
Tactic:
rw [inst.abs_imap, inst.abs_imap, inst.abs_imap, inst.abs_imap, inst.abs_imap, inst.abs_imap, h₂]
Proof state after:
no goals
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
⊢ ∀ (a b : Quot (congr F i)),
    r i a b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) a =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Tactic:
. intro x; apply Quot.inductionOn (motive := _) x; clear x
  intro x y; apply Quot.inductionOn (motive := _) y; clear y
  intro y h
  apply Quot.sound
  apply h
Proof state after:
no goals
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
⊢ ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b
Tactic:
intro a b h₃
Proof state after:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ Quot.mk (r' i) a = Quot.mk (r' i) b
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ Quot.mk (r' i) a = Quot.mk (r' i) b
Tactic:
apply Quot.sound
Proof state after:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r' i a b
Theorem: IQPF.M.bisim_lemma
Proof state before:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r' i a b
Tactic:
simp only
Proof state after:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r i (Quot.mk (congr F i) a) (Quot.mk (congr F i) b)
Theorem: IQPF.M.bisim_lemma
Proof state before:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r i (Quot.mk (congr F i) a) (Quot.mk (congr F i) b)
Tactic:
rw [Quot.sound h₃]
Proof state after:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r i (Quot.mk (congr F i) b) (Quot.mk (congr F i) b)
Theorem: IQPF.M.bisim_lemma
Proof state before:
case a
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
a b : IContainer.M (C F) i
h₃ : congr F i a b
⊢ r i (Quot.mk (congr F i) b) (Quot.mk (congr F i) b)
Tactic:
apply h₀
Proof state after:
no goals
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
⊢ ∀ (a b : Quot (congr F i)),
    r i a b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) a =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Tactic:
intro x
Proof state after:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝¹ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x✝ y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x✝ y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
x : Quot (congr F i)
⊢ ∀ (b : Quot (congr F i)),
    r i x b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) x =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝¹ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x✝ y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x✝ y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
x : Quot (congr F i)
⊢ ∀ (b : Quot (congr F i)),
    r i x b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) x =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Tactic:
apply Quot.inductionOn (motive := _) x
Proof state after:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝¹ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x✝ y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x✝ y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
x : Quot (congr F i)
⊢ ∀ (a : IContainer.M (C F) i) (b : Quot (congr F i)),
    r i (Quot.mk (congr F i) a) b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b)
          (Quot.mk (congr F i) a) =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Theorem: IQPF.M.bisim_lemma
Proof state before:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝¹ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x✝ y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x✝ y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
x : Quot (congr F i)
⊢ ∀ (a : IContainer.M (C F) i) (b : Quot (congr F i)),
    r i (Quot.mk (congr F i) a) b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b)
          (Quot.mk (congr F i) a) =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b
Tactic:
clear x
Proof state after:
I : Type u₁
F : (I → Type u₁) → I → Type u₁
inst : IQPF F
r : (i : I) → M F i → M F i → Prop
h₀ : ∀ (i : I) (x : M F i), r i x x
h₁ :
  ∀ (i : I) (x y : M F i),
    r i x y → IFunctor.imap (fun i => Quot.mk (r i)) (destruct x) = IFunctor.imap (fun i => Quot.mk (r i)) (destruct y)
i✝¹ : I
x✝ y✝ : IContainer.M (C F) i✝¹
r' : (i : I) → IContainer.M (C F) i → IContainer.M (C F) i → Prop :=
  fun i x y => r i (Quot.mk (congr F i) x) (Quot.mk (congr F i) y)
i✝ : I
x y : IContainer.M (C F) i✝
h₂✝ : r' i✝ x y
nx : IContainer.A (C F) i✝
kx : (y : IContainer.B (C F) i✝ nx) → IContainer.M (C F) (IContainer.N (C F) i✝ nx y)
ny : IContainer.A (C F) i✝
ky : (y : IContainer.B (C F) i✝ ny) → IContainer.M (C F) (IContainer.N (C F) i✝ ny y)
h₂ :
  IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := nx, snd := kx })) =
    IFunctor.imap (fun i => Quot.mk (r i))
      (IFunctor.imap (fun x x_1 => Quot.mk (congr F x) x_1) (abs { fst := ny, snd := ky }))
i : I
⊢ ∀ (a : IContainer.M (C F) i) (b : Quot (congr F i)),
    r i (Quot.mk (congr F i) a) b →
      Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b)
          (Quot.mk (congr F i) a) =
        Quot.lift (Quot.mk (r' i))
          (_ : ∀ (a b : IContainer.M (C F) i), congr F i a b → Quot.mk (r' i) a = Quot.mk (r' i) b) b