Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma0	[121, 1]	[131, 52]	apply HEq.symm	case h C : Container x : M C n : Nat i : B C (Approx.node (approx x (Nat.succ n))) j : B C (destruct x).fst heq : HEq i j this : B C (destruct x).fst = B C (Approx.node (approx x (Nat.succ n))) ⊢ HEq j i	case h.h C : Container x : M C n : Nat i : B C (Approx.node (approx x (Nat.succ n))) j : B C (destruct x).fst heq : HEq i j this : B C (destruct x).fst = B C (Approx.node (approx x (Nat.succ n))) ⊢ HEq i j	Please generate a tactic in lean4 to solve the state. STATE: case h C : Container x : M C n : Nat i : B C (Approx.node (approx x (Nat.succ n))) j : B C (destruct x).fst heq : HEq i j this : B C (destruct x).fst = B C (Approx.node (approx x (Nat.succ n))) ⊢ HEq j i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma0	[121, 1]	[131, 52]	assumption	case h.h C : Container x : M C n : Nat i : B C (Approx.node (approx x (Nat.succ n))) j : B C (destruct x).fst heq : HEq i j this : B C (destruct x).fst = B C (Approx.node (approx x (Nat.succ n))) ⊢ HEq i j	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h C : Container x : M C n : Nat i : B C (Approx.node (approx x (Nat.succ n))) j : B C (destruct x).fst heq : HEq i j this : B C (destruct x).fst = B C (Approx.node (approx x (Nat.succ n))) ⊢ HEq i j TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro x y n h₁	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) ⊢ ∀ (x y : M C) (n : Nat), R x y → approx x n = approx y n	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) x y : M C n : Nat h₁ : R x y ⊢ approx x n = approx y n	Please generate a tactic in lean4 to solve the state. STATE: 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) ⊢ ∀ (x y : M C) (n : Nat), R x y → approx x n = approx y n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	induction n generalizing x y	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) x y : M C n : Nat h₁ : R x y ⊢ approx x n = approx y n	case zero 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) x y : M C h₁ : R x y ⊢ approx x Nat.zero = approx y Nat.zero case succ 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 n_ih✝ : ∀ (x y : M C), R x y → approx x n✝ = approx y n✝ x y : M C h₁ : R x y ⊢ approx x (Nat.succ n✝) = approx y (Nat.succ n✝)	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C n : Nat h₁ : R x y ⊢ approx x n = approx y n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	case zero => cases x.approx 0 cases y.approx 0 rfl	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) x y : M C h₁ : R x y ⊢ approx x Nat.zero = approx y Nat.zero	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y ⊢ approx x Nat.zero = approx y Nat.zero TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	case succ n h => have h₂ := x.node_thm n have h₃ := y.node_thm n cases h₄: x.approx (.succ n) with \| MStep nodex cx => cases h₅: y.approx (.succ n) with \| MStep nodey cy => have ⟨node, k₁, k₂, eq₁, eq₂, kR⟩ := h₀ _ _ h₁ have h₂ : (Approx.MStep nodex cx).node = node := by have := congrArg (λ x => x.1) eq₁ simp only [destruct] at this rw [h₄] at h₂ rw [←this] exact h₂ have h₃ : (Approx.MStep nodey cy).node = node := by have := congrArg (λ x => x.1) eq₂ simp only [destruct] at this rw [h₅] at h₃ rw [←this] exact h₃ simp [Approx.node] at h₂ h₃ have h₂ := Eq.symm h₂ have h₃ := Eq.symm h₃ have h₆ := bisim.lemma0 x n have h₇ := bisim.lemma0 y n rw [h₄] at h₆ rw [h₅] at h₇ simp only [Approx.children] at h₇ h₆ induction h₂ induction h₃ suffices h₈ : ∀ i: C.B node, cx i = cy i by have : cx = cy := by funext i apply h₈ rw [this] intro i have h₆ : ∀ i, cx i = (k₁ i).approx n := by have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by rw [eq₁] intro i j heq cases heq rfl intro i rw [h₆ i (cast (by simp [congrArg (λ x => x.1) eq₁]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq have h₇ : ∀ i, cy i = (k₂ i).approx n := by have h : ∀ i j, HEq i j → (y.destruct.snd i) = k₂ j := by rw [eq₂] intro i j heq cases heq rfl intro i rw [h₇ i (cast (by simp [congrArg (λ x => x.1) eq₂]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq rw [h₆, h₇] apply h apply kR	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 ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ approx x (Nat.succ n) = approx y (Nat.succ n) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases x.approx 0	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) x y : M C h₁ : R x y ⊢ approx x Nat.zero = approx y Nat.zero	case MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = approx y Nat.zero	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y ⊢ approx x Nat.zero = approx y Nat.zero TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases y.approx 0	case MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = approx y Nat.zero	case MStop.MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = Approx.MStop	Please generate a tactic in lean4 to solve the state. STATE: case MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = approx y Nat.zero TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rfl	case MStop.MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = Approx.MStop	no goals	Please generate a tactic in lean4 to solve the state. STATE: case MStop.MStop 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) x y : M C h₁ : R x y ⊢ Approx.MStop = Approx.MStop TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₂ := x.node_thm n	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 ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	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)) = node x ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ approx x (Nat.succ n) = approx y (Nat.succ n) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₃ := y.node_thm n	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)) = node x ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = node y ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	Please generate a tactic in lean4 to solve the state. STATE: 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)) = node x ⊢ approx x (Nat.succ n) = approx y (Nat.succ n) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases h₄: x.approx (.succ n) with \| MStep nodex cx => cases h₅: y.approx (.succ n) with \| MStep nodey cy => have ⟨node, k₁, k₂, eq₁, eq₂, kR⟩ := h₀ _ _ h₁ have h₂ : (Approx.MStep nodex cx).node = node := by have := congrArg (λ x => x.1) eq₁ simp only [destruct] at this rw [h₄] at h₂ rw [←this] exact h₂ have h₃ : (Approx.MStep nodey cy).node = node := by have := congrArg (λ x => x.1) eq₂ simp only [destruct] at this rw [h₅] at h₃ rw [←this] exact h₃ simp [Approx.node] at h₂ h₃ have h₂ := Eq.symm h₂ have h₃ := Eq.symm h₃ have h₆ := bisim.lemma0 x n have h₇ := bisim.lemma0 y n rw [h₄] at h₆ rw [h₅] at h₇ simp only [Approx.children] at h₇ h₆ induction h₂ induction h₃ suffices h₈ : ∀ i: C.B node, cx i = cy i by have : cx = cy := by funext i apply h₈ rw [this] intro i have h₆ : ∀ i, cx i = (k₁ i).approx n := by have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by rw [eq₁] intro i j heq cases heq rfl intro i rw [h₆ i (cast (by simp [congrArg (λ x => x.1) eq₁]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq have h₇ : ∀ i, cy i = (k₂ i).approx n := by have h : ∀ i j, HEq i j → (y.destruct.snd i) = k₂ j := by rw [eq₂] intro i j heq cases heq rfl intro i rw [h₇ i (cast (by simp [congrArg (λ x => x.1) eq₂]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq rw [h₆, h₇] apply h apply kR	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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = node y ⊢ approx x (Nat.succ n) = approx y (Nat.succ n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = node y ⊢ approx x (Nat.succ n) = approx y (Nat.succ n) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases h₅: y.approx (.succ n) with \| MStep nodey cy => have ⟨node, k₁, k₂, eq₁, eq₂, kR⟩ := h₀ _ _ h₁ have h₂ : (Approx.MStep nodex cx).node = node := by have := congrArg (λ x => x.1) eq₁ simp only [destruct] at this rw [h₄] at h₂ rw [←this] exact h₂ have h₃ : (Approx.MStep nodey cy).node = node := by have := congrArg (λ x => x.1) eq₂ simp only [destruct] at this rw [h₅] at h₃ rw [←this] exact h₃ simp [Approx.node] at h₂ h₃ have h₂ := Eq.symm h₂ have h₃ := Eq.symm h₃ have h₆ := bisim.lemma0 x n have h₇ := bisim.lemma0 y n rw [h₄] at h₆ rw [h₅] at h₇ simp only [Approx.children] at h₇ h₆ induction h₂ induction h₃ suffices h₈ : ∀ i: C.B node, cx i = cy i by have : cx = cy := by funext i apply h₈ rw [this] intro i have h₆ : ∀ i, cx i = (k₁ i).approx n := by have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by rw [eq₁] intro i j heq cases heq rfl intro i rw [h₆ i (cast (by simp [congrArg (λ x => x.1) eq₁]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq have h₇ : ∀ i, cy i = (k₂ i).approx n := by have h : ∀ i j, HEq i j → (y.destruct.snd i) = k₂ j := by rw [eq₂] intro i j heq cases heq rfl intro i rw [h₇ i (cast (by simp [congrArg (λ x => x.1) eq₂]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq rw [h₆, h₇] apply h apply kR	case MStep 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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = node y nodex : C.A cx : B C nodex → Approx C n h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx ⊢ Approx.MStep nodex cx = approx y (Nat.succ n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case MStep 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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = node y nodex : C.A cx : B C nodex → Approx C n h₄ : approx x (Nat.succ n) = Approx.MStep nodex cx ⊢ Approx.MStep nodex cx = approx y (Nat.succ n) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have ⟨node, k₁, k₂, eq₁, eq₂, kR⟩ := h₀ _ _ h₁	case MStep.MStep 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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = 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 ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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) ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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)) = node x h₃ : Approx.node (approx y (Nat.succ n)) = 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 ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₂ : (Approx.MStep nodex cx).node = node := by have := congrArg (λ x => x.1) eq₁ simp only [destruct] at this rw [h₄] at h₂ rw [←this] exact h₂	case MStep.MStep 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) ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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 ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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) ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₃ : (Approx.MStep nodey cy).node = node := by have := congrArg (λ x => x.1) eq₂ simp only [destruct] at this rw [h₅] at h₃ rw [←this] exact h₃	case MStep.MStep 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 ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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 h₃ : Approx.node (Approx.MStep nodey cy) = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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 ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp [Approx.node] at h₂ h₃	case MStep.MStep 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 h₃ : Approx.node (Approx.MStep nodey cy) = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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 : nodex = node h₃ : nodey = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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 h₃ : Approx.node (Approx.MStep nodey cy) = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₂ := Eq.symm h₂	case MStep.MStep 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 : nodex = node h₃ : nodey = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : 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✝ : nodex = node h₃ : nodey = node h₂ : node = nodex ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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 : nodex = node h₃ : nodey = node ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₃ := Eq.symm h₃	case MStep.MStep 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✝¹ : 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✝ : nodex = node h₃ : nodey = node h₂ : node = nodex ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : 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✝ : nodex = node h₃ : nodey = node h₂ : node = nodex ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₆ := bisim.lemma0 x n	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₇ := bisim.lemma0 y n	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₄] at h₆	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (approx x (Nat.succ n)))) (j : B C (destruct x).fst), HEq i j → Approx.children (approx x (Nat.succ n)) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₅] at h₇	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → Approx.children (Approx.MStep nodey cy) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (approx y (Nat.succ n)))) (j : B C (destruct y).fst), HEq i j → Approx.children (approx y (Nat.succ n)) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp only [Approx.children] at h₇ h₆	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → Approx.children (Approx.MStep nodey cy) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → cx i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → Approx.children (Approx.MStep nodex cx) i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → Approx.children (Approx.MStep nodey cy) i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	induction h₂	case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → cx i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy	case MStep.MStep.refl 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✝¹ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex 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) h₃✝ : nodey = node h₃ : node = nodey h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n 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 ⊢ Approx.MStep node cx = Approx.MStep nodey cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep 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✝¹ : Approx.node (approx x (Nat.succ n)) = Container.M.node x h₃✝¹ : 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✝ : nodex = node h₃✝ : nodey = node h₂ : node = nodex h₃ : node = nodey h₆ : ∀ (i : B C (Approx.node (Approx.MStep nodex cx))) (j : B C (destruct x).fst), HEq i j → cx i = approx (PSigma.snd (destruct x) j) n h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n ⊢ Approx.MStep nodex cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	induction h₃	case MStep.MStep.refl 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✝¹ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex 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) h₃✝ : nodey = node h₃ : node = nodey h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n 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 ⊢ Approx.MStep node cx = Approx.MStep nodey cy	case MStep.MStep.refl.refl 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 ⊢ Approx.MStep node cx = Approx.MStep node cy	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl 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✝¹ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex 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) h₃✝ : nodey = node h₃ : node = nodey h₇ : ∀ (i : B C (Approx.node (Approx.MStep nodey cy))) (j : B C (destruct y).fst), HEq i j → cy i = approx (PSigma.snd (destruct y) j) n 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 ⊢ Approx.MStep node cx = Approx.MStep nodey cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	suffices h₈ : ∀ i: C.B node, cx i = cy i by have : cx = cy := by funext i apply h₈ rw [this]	case MStep.MStep.refl.refl 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 ⊢ Approx.MStep node cx = Approx.MStep node cy	case MStep.MStep.refl.refl 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), cx i = cy i	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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 ⊢ Approx.MStep node cx = Approx.MStep node cy TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro i	case MStep.MStep.refl.refl 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), cx i = cy i	case MStep.MStep.refl.refl 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 ⊢ cx i = cy i	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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), cx i = cy i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₆ : ∀ i, cx i = (k₁ i).approx n := by have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by rw [eq₁] intro i j heq cases heq rfl intro i rw [h₆ i (cast (by simp [congrArg (λ x => x.1) eq₁]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq	case MStep.MStep.refl.refl 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 ⊢ cx i = cy i	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ cx i = cy i	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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 ⊢ cx i = cy i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h₇ : ∀ i, cy i = (k₂ i).approx n := by have h : ∀ i j, HEq i j → (y.destruct.snd i) = k₂ j := by rw [eq₂] intro i j heq cases heq rfl intro i rw [h₇ i (cast (by simp [congrArg (λ x => x.1) eq₂]) i)] . rw [h] apply cast_heq . apply HEq.symm apply cast_heq	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ cx i = cy i	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ cx i = cy i	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ cx i = cy i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₆, h₇]	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ cx i = cy i	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ approx (k₁ i) n = approx (k₂ i) n	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ cx i = cy i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply h	case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ approx (k₁ i) n = approx (k₂ i) n	case MStep.MStep.refl.refl.h₁ 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ R (k₁ i) (k₂ i)	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ approx (k₁ i) n = approx (k₂ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply kR	case MStep.MStep.refl.refl.h₁ 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ R (k₁ i) (k₂ i)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case MStep.MStep.refl.refl.h₁ 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h₇ : ∀ (i : B C node), cy i = approx (k₂ i) n ⊢ R (k₁ i) (k₂ i) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have := congrArg (λ x => x.1) eq₁	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) ⊢ Approx.node (Approx.MStep nodex cx) = node	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) this : (fun x => x.fst) (destruct x) = (fun x => x.fst) { fst := node, snd := k₁ } ⊢ Approx.node (Approx.MStep nodex cx) = node	Please generate a tactic in lean4 to solve the state. STATE: 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) ⊢ Approx.node (Approx.MStep nodex cx) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp only [destruct] at this	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) this : (fun x => x.fst) (destruct x) = (fun x => x.fst) { fst := node, snd := k₁ } ⊢ Approx.node (Approx.MStep nodex cx) = node	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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node	Please generate a tactic in lean4 to solve the state. STATE: 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) this : (fun x => x.fst) (destruct x) = (fun x => x.fst) { fst := node, snd := k₁ } ⊢ Approx.node (Approx.MStep nodex cx) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₄] at h₂	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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node	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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node	Please generate a tactic in lean4 to solve the state. STATE: 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [←this]	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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node	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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = Container.M.node x	Please generate a tactic in lean4 to solve the state. STATE: 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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	exact h₂	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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = Container.M.node x	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 h₃ : Approx.node (approx y (Nat.succ n)) = Container.M.node y nodex : C.A cx : B C nodex → Approx C n h₂ : Approx.node (Approx.MStep nodex cx) = Container.M.node x 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) this : Container.M.node x = node ⊢ Approx.node (Approx.MStep nodex cx) = Container.M.node x TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have := congrArg (λ x => x.1) eq₂	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 ⊢ Approx.node (Approx.MStep nodey cy) = node	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 : (fun x => x.fst) (destruct y) = (fun x => x.fst) { fst := node, snd := k₂ } ⊢ Approx.node (Approx.MStep nodey cy) = node	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ Approx.node (Approx.MStep nodey cy) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp only [destruct] at this	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 : (fun x => x.fst) (destruct y) = (fun x => x.fst) { fst := node, snd := k₂ } ⊢ Approx.node (Approx.MStep nodey cy) = node	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	Please generate a tactic in lean4 to solve the state. STATE: 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 : (fun x => x.fst) (destruct y) = (fun x => x.fst) { fst := node, snd := k₂ } ⊢ Approx.node (Approx.MStep nodey cy) = node TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₅] at h₃	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [←this]	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	exact h₃	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	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have : cx = cy := by funext i apply h₈	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [this]	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	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	funext i	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply h₈	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	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h : ∀ i j, HEq i j → (x.destruct.snd i) = k₁ j := by rw [eq₁] intro i j heq cases heq rfl	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro i	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₆ i (cast (by simp [congrArg (λ x => x.1) eq₁]) i)]	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	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 ⊢ approx (PSigma.snd (destruct x) (cast (_ : B C node = B C (destruct x).fst) i)) n = approx (k₁ i) n 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)	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	. rw [h] apply cast_heq	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 ⊢ approx (PSigma.snd (destruct x) (cast (_ : B C node = B C (destruct x).fst) i)) n = approx (k₁ i) n 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)	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)	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ approx (PSigma.snd (destruct x) (cast (_ : B C node = B C (destruct x).fst) i)) n = approx (k₁ i) n 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	. apply HEq.symm apply cast_heq	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)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [eq₁]	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro i j heq	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	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	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases heq	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	case refl 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 ⊢ PSigma.snd { fst := node, snd := k₁ } i = k₁ i	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rfl	case refl 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 ⊢ PSigma.snd { fst := node, snd := k₁ } i = k₁ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refl 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 ⊢ PSigma.snd { fst := node, snd := k₁ } i = k₁ i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp [congrArg (λ x => x.1) eq₁]	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 ⊢ B C node = B C (destruct x).fst	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ B C node = B C (destruct x).fst TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [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 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 ⊢ approx (PSigma.snd (destruct x) (cast (_ : B C node = B C (destruct x).fst) i)) n = approx (k₁ i) n	case a 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 (cast (_ : B C node = B C (destruct x).fst) i) i	Please generate a tactic in lean4 to solve the state. STATE: 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 ⊢ approx (PSigma.snd (destruct x) (cast (_ : B C node = B C (destruct x).fst) i)) n = approx (k₁ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply cast_heq	case a 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 (cast (_ : B C node = B C (destruct x).fst) i) i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a 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 (cast (_ : B C node = B C (destruct x).fst) i) i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply HEq.symm	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)	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 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 (cast (_ : B C node = B C (destruct x).fst) i) i	Please generate a tactic in lean4 to solve the state. STATE: 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:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply cast_heq	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 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 (cast (_ : B C node = B C (destruct x).fst) i) i	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 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 (cast (_ : B C node = B C (destruct x).fst) i) i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	have h : ∀ i j, HEq i j → (y.destruct.snd i) = k₂ j := by rw [eq₂] intro i j heq cases heq rfl	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C node), cy i = approx (k₂ i) n	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j ⊢ ∀ (i : B C node), cy i = approx (k₂ i) n	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C node), cy i = approx (k₂ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro i	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j ⊢ ∀ (i : B C node), cy i = approx (k₂ i) n	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ cy i = approx (k₂ i) n	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j ⊢ ∀ (i : B C node), cy i = approx (k₂ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [h₇ i (cast (by simp [congrArg (λ x => x.1) eq₂]) i)]	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ cy i = approx (k₂ i) n	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ approx (PSigma.snd (destruct y) (cast (_ : B C node = B C (destruct y).fst) i)) n = approx (k₂ i) n 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i)	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ cy i = approx (k₂ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	. rw [h] apply cast_heq	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ approx (PSigma.snd (destruct y) (cast (_ : B C node = B C (destruct y).fst) i)) n = approx (k₂ i) n 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i)	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i)	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ approx (PSigma.snd (destruct y) (cast (_ : B C node = B C (destruct y).fst) i)) n = approx (k₂ i) n 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	. apply HEq.symm apply cast_heq	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [eq₂]	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C { fst := node, snd := k₂ }.fst) (j : B C node), HEq i j → PSigma.snd { fst := node, snd := k₂ } i = k₂ j	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	intro i j heq	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C { fst := node, snd := k₂ }.fst) (j : B C node), HEq i j → PSigma.snd { fst := node, snd := k₂ } i = k₂ j	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst j : B C node heq : HEq i j ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ j	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n ⊢ ∀ (i : B C { fst := node, snd := k₂ }.fst) (j : B C node), HEq i j → PSigma.snd { fst := node, snd := k₂ } i = k₂ j TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	cases heq	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst j : B C node heq : HEq i j ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ j	case refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ i	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst j : B C node heq : HEq i j ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ j TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rfl	case refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refl 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n i : B C { fst := node, snd := k₂ }.fst ⊢ PSigma.snd { fst := node, snd := k₂ } i = k₂ i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	simp [congrArg (λ x => x.1) eq₂]	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ B C node = B C (destruct y).fst	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ B C node = B C (destruct y).fst TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	rw [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 i✝ : B C node h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ approx (PSigma.snd (destruct y) (cast (_ : B C node = B C (destruct y).fst) i)) n = approx (k₂ i) n	case a 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ approx (PSigma.snd (destruct y) (cast (_ : B C node = B C (destruct y).fst) i)) n = approx (k₂ i) n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply cast_heq	case a 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply HEq.symm	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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i)	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 i✝ : B C node h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i	Please generate a tactic in lean4 to solve the state. STATE: 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 h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq i (cast (_ : B C node = B C (destruct y).fst) i) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim.lemma1	[136, 1]	[219, 17]	apply cast_heq	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 i✝ : B C node h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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 i✝ : B C node h₆ : ∀ (i : B C node), cx i = approx (k₁ i) n h : ∀ (i : B C (destruct y).fst) (j : B C node), HEq i j → PSigma.snd (destruct y) i = k₂ j i : B C node ⊢ HEq (cast (_ : B C node = B C (destruct y).fst) i) i TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	intro x y h₁	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) ⊢ ∀ (x y : M C), R x y → x = y	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) x y : M C h₁ : R x y ⊢ x = y	Please generate a tactic in lean4 to solve the state. STATE: 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) ⊢ ∀ (x y : M C), R x y → x = y TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	suffices h₂: x.approx = y.approx by cases x cases y simp only at h₂ simp [h₂]	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) x y : M C h₁ : R x y ⊢ x = y	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) x y : M C h₁ : R x y ⊢ x.approx = y.approx	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y ⊢ x = y TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	funext n	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) x y : M C h₁ : R x y ⊢ x.approx = y.approx	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) x y : M C h₁ : R x y n : Nat ⊢ approx x n = approx y n	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y ⊢ x.approx = y.approx TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	apply bisim.lemma1 R h₀ x y n h₁	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) x y : M C h₁ : R x y n : Nat ⊢ approx x n = approx y n	no goals	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y n : Nat ⊢ approx x n = approx y n TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	cases x	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) x y : M C h₁ : R x y h₂ : x.approx = y.approx ⊢ x = y	case mk 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) y : M C approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝, agrees := agrees✝ } y h₂ : { approx := approx✝, agrees := agrees✝ }.approx = y.approx ⊢ { approx := approx✝, agrees := agrees✝ } = y	Please generate a tactic in lean4 to solve the state. STATE: 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) x y : M C h₁ : R x y h₂ : x.approx = y.approx ⊢ x = y TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	cases y	case mk 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) y : M C approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝, agrees := agrees✝ } y h₂ : { approx := approx✝, agrees := agrees✝ }.approx = y.approx ⊢ { approx := approx✝, agrees := agrees✝ } = y	case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : { approx := approx✝¹, agrees := agrees✝¹ }.approx = { approx := approx✝, agrees := agrees✝ }.approx ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ }	Please generate a tactic in lean4 to solve the state. STATE: case mk 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) y : M C approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝, agrees := agrees✝ } y h₂ : { approx := approx✝, agrees := agrees✝ }.approx = y.approx ⊢ { approx := approx✝, agrees := agrees✝ } = y TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	simp only at h₂	case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : { approx := approx✝¹, agrees := agrees✝¹ }.approx = { approx := approx✝, agrees := agrees✝ }.approx ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ }	case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : approx✝¹ = approx✝ ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ }	Please generate a tactic in lean4 to solve the state. STATE: case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : { approx := approx✝¹, agrees := agrees✝¹ }.approx = { approx := approx✝, agrees := agrees✝ }.approx ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ } TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/M.lean	Container.M.bisim	[221, 1]	[232, 35]	simp [h₂]	case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : approx✝¹ = approx✝ ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ }	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mk.mk 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) approx✝¹ : (n : Nat) → Approx C n agrees✝¹ : ∀ (n : Nat), Agree (approx✝¹ n) (approx✝¹ (Nat.succ n)) approx✝ : (n : Nat) → Approx C n agrees✝ : ∀ (n : Nat), Agree (approx✝ n) (approx✝ (Nat.succ n)) h₁ : R { approx := approx✝¹, agrees := agrees✝¹ } { approx := approx✝, agrees := agrees✝ } h₂ : approx✝¹ = approx✝ ⊢ { approx := approx✝¹, agrees := agrees✝¹ } = { approx := approx✝, agrees := agrees✝ } TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.by_cases	[106, 15]	[113, 17]	intro x	F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) ⊢ (x : Free F R) → motive x	F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) ⊢ (x : Free F R) → motive x TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.by_cases	[106, 15]	[113, 17]	rw [←construct_destruct x]	F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive x	F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive (construct (destruct x))	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive x TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.by_cases	[106, 15]	[113, 17]	cases destruct x with \| Pure r => exact pure r \| Free f => exact free f	F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive (construct (destruct x))	no goals	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R ⊢ motive (construct (destruct x)) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.by_cases	[106, 15]	[113, 17]	exact pure r	case Pure F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R r : R ⊢ motive (construct (FreeF.Pure r))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case Pure F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R r : R ⊢ motive (construct (FreeF.Pure r)) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.by_cases	[106, 15]	[113, 17]	exact free f	case Free F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R f : F (Free F R) ⊢ motive (construct (FreeF.Free f))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case Free F : Type u → Type u inst : QPF F R : Type u motive : Free F R → Sort u_1 pure : (r : R) → motive (Pure.pure r) free : (f : F (Free F R)) → motive (Free.free f) x : Free F R f : F (Free F R) ⊢ motive (construct (FreeF.Free f)) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.destruct_pure	[115, 9]	[116, 34]	simp [pure, destruct_construct]	F : Type u → Type u inst : QPF F R : Type u r : R ⊢ destruct (pure r) = FreeF.Pure r	no goals	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R : Type u r : R ⊢ destruct (pure r) = FreeF.Pure r TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.destruct_free	[118, 9]	[119, 34]	simp [free, destruct_construct]	F : Type u → Type u inst : QPF F R : Type u f : F (Free F R) ⊢ destruct (free f) = FreeF.Free f	no goals	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R : Type u f : F (Free F R) ⊢ destruct (free f) = FreeF.Free f TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	apply QPF.M.bisim (λ x y => corec (bind.automaton k) (.inr x) = y)	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : Free F S), corec (bind.automaton k) (Sum.inr x) = y → x = y	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : QPF.M (FreeF F S)), corec (bind.automaton k) (Sum.inr x) = y → QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y)	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : Free F S), corec (bind.automaton k) (Sum.inr x) = y → x = y TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	. intro x y h₁ induction h₁ rw [corec, QPF.M.destruct_corec] conv => congr . skip . skip . rhs simp only [bind.automaton] exists (λ x => ⟨⟨x, corec (bind.automaton k) (.inr x)⟩, by rfl⟩) <$> destruct x rw [destruct] constructor . simp only [←QPF.map_comp, Function.comp] apply Eq.trans _ (QPF.map_id _) rfl . simp only [←QPF.map_comp, Function.comp] rfl	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : QPF.M (FreeF F S)), corec (bind.automaton k) (Sum.inr x) = y → QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y)	no goals	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : QPF.M (FreeF F S)), corec (bind.automaton k) (Sum.inr x) = y → QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	intro x y h₁	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : QPF.M (FreeF F S)), corec (bind.automaton k) (Sum.inr x) = y → QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y)	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) h₁ : corec (bind.automaton k) (Sum.inr x) = y ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y)	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S ⊢ ∀ (x y : QPF.M (FreeF F S)), corec (bind.automaton k) (Sum.inr x) = y → QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	induction h₁	F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) h₁ : corec (bind.automaton k) (Sum.inr x) = y ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y)	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct (corec (bind.automaton k) (Sum.inr x)))	Please generate a tactic in lean4 to solve the state. STATE: F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) h₁ : corec (bind.automaton k) (Sum.inr x) = y ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct y) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	rw [corec, QPF.M.destruct_corec]	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct (corec (bind.automaton k) (Sum.inr x)))	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> bind.automaton k (Sum.inr x))	Please generate a tactic in lean4 to solve the state. STATE: case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) (QPF.M.destruct (corec (bind.automaton k) (Sum.inr x))) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	conv => congr . skip . skip . rhs simp only [bind.automaton]	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> bind.automaton k (Sum.inr x))	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x)	Please generate a tactic in lean4 to solve the state. STATE: case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> bind.automaton k (Sum.inr x)) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	exists (λ x => ⟨⟨x, corec (bind.automaton k) (.inr x)⟩, by rfl⟩) <$> destruct x	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x)	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x	Please generate a tactic in lean4 to solve the state. STATE: case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ QPF.M.liftr (FreeF F S) (fun x y => QPF.M.corec (bind.automaton k) (Sum.inr x) = y) (QPF.M.destruct x) ((fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x) TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	rw [destruct]	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x	Please generate a tactic in lean4 to solve the state. STATE: case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> destruct x TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	constructor	case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x	case refl.left F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x case refl.right F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x	Please generate a tactic in lean4 to solve the state. STATE: case refl F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x ∧ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x TACTIC:
https://github.com/RemyCiterin/LeanCoInd.git	69d305ae769624f460f9c1ee6a0351917f4b74cf	CoInd/QPF/FreeMonads.lean	Free.bind_inr.internal	[186, 1]	[205, 10]	. simp only [←QPF.map_comp, Function.comp] apply Eq.trans _ (QPF.map_id _) rfl	case refl.left F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x case refl.right F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x	case refl.right F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x	Please generate a tactic in lean4 to solve the state. STATE: case refl.left F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).fst) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = QPF.M.destruct x case refl.right F : Type u → Type u inst : QPF F R S : Type u k : R → Free F S x y : QPF.M (FreeF F S) ⊢ (fun x => (↑x).snd) <$> (fun x => { val := (x, corec (bind.automaton k) (Sum.inr x)), property := (_ : QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst) = QPF.M.corec (bind.automaton k) (Sum.inr (x, corec (bind.automaton k) (Sum.inr x)).fst)) }) <$> QPF.M.destruct x = (fun x => QPF.M.corec (bind.automaton k) x) <$> Sum.inr <$> QPF.M.destruct x TACTIC:

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