internlm/Lean-Github huggingface dataset:
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex : C.A
cx : B C nodex → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx
nodey : C.A
cy : B C nodey → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep nodey cy
node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
h₂ : Approx.node (Approx.MStep nodex cx) = node
this : Container.M.node y = node
⊢ Approx.node (Approx.MStep nodey cy) = node
Tactic:
rw [h₅] at h₃
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
nodex : C.A
cx : B C nodex → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx
nodey : C.A
cy : B C nodey → Approx C n
h₃ : Approx.node (Approx.MStep nodey cy) = Container.M.node y
h₅ : approx y (Nat.succ n) = Approx.MStep nodey cy
node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
h₂ : Approx.node (Approx.MStep nodex cx) = node
this : Container.M.node y = node
⊢ Approx.node (Approx.MStep nodey cy) = node
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
nodex : C.A
cx : B C nodex → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx
nodey : C.A
cy : B C nodey → Approx C n
h₃ : Approx.node (Approx.MStep nodey cy) = Container.M.node y
h₅ : approx y (Nat.succ n) = Approx.MStep nodey cy
node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
h₂ : Approx.node (Approx.MStep nodex cx) = node
this : Container.M.node y = node
⊢ Approx.node (Approx.MStep nodey cy) = node
Tactic:
rw [←this]
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
nodex : C.A
cx : B C nodex → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx
nodey : C.A
cy : B C nodey → Approx C n
h₃ : Approx.node (Approx.MStep nodey cy) = Container.M.node y
h₅ : approx y (Nat.succ n) = Approx.MStep nodey cy
node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
h₂ : Approx.node (Approx.MStep nodex cx) = node
this : Container.M.node y = node
⊢ Approx.node (Approx.MStep nodey cy) = Container.M.node y
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
nodex : C.A
cx : B C nodex → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx
nodey : C.A
cy : B C nodey → Approx C n
h₃ : Approx.node (Approx.MStep nodey cy) = Container.M.node y
h₅ : approx y (Nat.succ n) = Approx.MStep nodey cy
node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
h₂ : Approx.node (Approx.MStep nodex cx) = node
this : Container.M.node y = node
⊢ Approx.node (Approx.MStep nodey cy) = Container.M.node y
Tactic:
exact h₃
Proof state after:
no goals
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
⊢ Approx.MStep node cx = Approx.MStep node cy
Tactic:
have : cx = cy := by
  funext i
  apply h₈
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
this : cx = cy
⊢ Approx.MStep node cx = Approx.MStep node cy
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
this : cx = cy
⊢ Approx.MStep node cx = Approx.MStep node cy
Tactic:
rw [this]
Proof state after:
no goals
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
⊢ cx = cy
Tactic:
funext i
Proof state after:
case h
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
i : B C node
⊢ cx i = cy i
Theorem: Container.M.bisim.lemma1
Proof state before:
case h
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
h₈ : ∀ (i : B C node), cx i = cy i
i : B C node
⊢ cx i = cy i
Tactic:
apply h₈
Proof state after:
no goals
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
⊢ ∀ (i : B C node), cx i = approx (k₁ i) n
Tactic:
have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by
  rw [eq₁]
  intro i j heq
  cases heq
  rfl
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h✝ : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
h : ∀ (i : B C (destruct x).fst) (j : B C node), HEq i j → PSigma.snd (destruct x) i = k₁ j
⊢ ∀ (i : B C node), cx i = approx (k₁ i) n
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h✝ : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
h : ∀ (i : B C (destruct x).fst) (j : B C node), HEq i j → PSigma.snd (destruct x) i = k₁ j
⊢ ∀ (i : B C node), cx i = approx (k₁ i) n
Tactic:
intro i
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h✝ : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i✝ : B C node
h : ∀ (i : B C (destruct x).fst) (j : B C node), HEq i j → PSigma.snd (destruct x) i = k₁ j
i : B C node
⊢ cx i = approx (k₁ i) n
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h✝ : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i✝ : B C node
h : ∀ (i : B C (destruct x).fst) (j : B C node), HEq i j → PSigma.snd (destruct x) i = k₁ j
i : B C node
⊢ HEq i (cast (_ : B C node = B C (destruct x).fst) i)
Tactic:
. apply HEq.symm
  apply cast_heq
Proof state after:
no goals
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
⊢ ∀ (i : B C (destruct x).fst) (j : B C node), HEq i j → PSigma.snd (destruct x) i = k₁ j
Tactic:
rw [eq₁]
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
⊢ ∀ (i : B C { fst := node, snd := k₁ }.fst) (j : B C node), HEq i j → PSigma.snd { fst := node, snd := k₁ } i = k₁ j
Theorem: Container.M.bisim.lemma1
Proof state before:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i : B C node
⊢ ∀ (i : B C { fst := node, snd := k₁ }.fst) (j : B C node), HEq i j → PSigma.snd { fst := node, snd := k₁ } i = k₁ j
Tactic:
intro i j heq
Proof state after:
C : Container
R : M C → M C → Prop
h₀ :
  ∀ (x y : M C),
    R x y →
      ∃ node k₁ k₂,
        destruct x = { fst := node, snd := k₁ } ∧
          destruct y = { fst := node, snd := k₂ } ∧ ∀ (i : B C node), R (k₁ i) (k₂ i)
n : Nat
h : ∀ (x y : M C), R x y → approx x n = approx y n
x y : M C
h₁ : R x y
h₂✝ : Approx.node (approx x (Nat.succ n)) = Container.M.node x
h₃✝ : Approx.node (approx y (Nat.succ n)) = Container.M.node y
nodex nodey node : C.A
k₁ k₂ : B C node → M C
eq₁ : destruct x = { fst := node, snd := k₁ }
eq₂ : destruct y = { fst := node, snd := k₂ }
kR : ∀ (i : B C node), R (k₁ i) (k₂ i)
cx : B C node → Approx C n
h₄ : approx x (Nat.succ n) = Approx.MStep node cx
h₂ : node = node
h₆ :
  ∀ (i : B C (Approx.node (Approx.MStep node cx))) (j : B C (destruct x).fst),
    HEq i j → cx i = approx (PSigma.snd (destruct x) j) n
cy : B C node → Approx C n
h₅ : approx y (Nat.succ n) = Approx.MStep node cy
h₃ : node = node
h₇ :
  ∀ (i : B C (Approx.node (Approx.MStep node cy))) (j : B C (destruct y).fst),
    HEq i j → cy i = approx (PSigma.snd (destruct y) j) n
i✝ : B C node
i : B C { fst := node, snd := k₁ }.fst
j : B C node
heq : HEq i j
⊢ PSigma.snd { fst := node, snd := k₁ } i = k₁ j