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/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	right	case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ ?inl.inr.left.a.b = X.b ∨ ?inl.inr.left.a.b = X.d case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inl.inr.left.a.b X.d case inl.inr.left.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ X	case inl.inr.left.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ ?inl.inr.left.a.b = X.d case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inl.inr.left.a.b X.d case inl.inr.left.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ ?inl.inr.left.a.b = X.b ∨ ?inl.inr.left.a.b = X.d case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inl.inr.left.a.b X.d case inl.inr.left.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case inl.inr.left.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ ?inl.inr.left.a.b = X.d case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inl.inr.left.a.b X.d case inl.inr.left.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ X	case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	Please generate a tactic in lean4 to solve the state. STATE: case inl.inr.left.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ ?inl.inr.left.a.b = X.d case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inl.inr.left.a.b X.d case inl.inr.left.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inl.inr.left.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inl.inr.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inl.inr.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.b h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	exists X.d	case inr.inl A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ joins (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.b	case inr.inl A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ joins (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.b TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inl A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	case inr.inl.left A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. constructor	case inr.inl.left A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.left A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. constructor; simp [RX]; right; trivial constructor; simp [RX]; right; trivial constructor	case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inl.left A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.left A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.b = X.b ∧ ?inr.inl.right.b = X.a ∨ X.b = X.b ∧ ?inr.inl.right.b = X.c ∨ X.b = X.c ∧ ?inr.inl.right.b = X.b ∨ X.b = X.c ∧ ?inr.inl.right.b = X.d case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	simp [RX]	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.b = X.b ∧ ?inr.inl.right.b = X.a ∨ X.b = X.b ∧ ?inr.inl.right.b = X.c ∨ X.b = X.c ∧ ?inr.inl.right.b = X.b ∨ X.b = X.c ∧ ?inr.inl.right.b = X.d case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.a ∨ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.b = X.b ∧ ?inr.inl.right.b = X.a ∨ X.b = X.b ∧ ?inr.inl.right.b = X.c ∨ X.b = X.c ∧ ?inr.inl.right.b = X.b ∨ X.b = X.c ∧ ?inr.inl.right.b = X.d case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	right	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.a ∨ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.a ∨ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case inr.inl.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.c X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.b = X.c case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.b X.d case inr.inl.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.c X.d	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.c = X.b ∧ ?inr.inl.right.a.b = X.a ∨ X.c = X.b ∧ ?inr.inl.right.a.b = X.c ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.b ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.c X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	simp [RX]	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.c = X.b ∧ ?inr.inl.right.a.b = X.a ∨ X.c = X.b ∧ ?inr.inl.right.a.b = X.c ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.b ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.b ∨ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X.c = X.b ∧ ?inr.inl.right.a.b = X.a ∨ X.c = X.b ∧ ?inr.inl.right.a.b = X.c ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.b ∨ X.c = X.c ∧ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	right	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.b ∨ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.b ∨ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case inr.inl.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ ?inr.inl.right.a.b = X.d case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) ?inr.inl.right.a.b X.d case inr.inl.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.inl.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.b ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	exists X.d	case inr.inr A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ joins (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	case inr.inr A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	Please generate a tactic in lean4 to solve the state. STATE: case inr.inr A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ joins (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	repeat constructor	case inr.inr A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.inr A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d ∧ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case inr.inr.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.right A : Type R : A → A → Prop inhabited_A : Nonempty A x y z : X h✝¹ : y = X.d h✝ : z = X.d ⊢ refl_trans_clos (fun x y => x = X.b ∧ y = X.a ∨ x = X.b ∧ y = X.c ∨ x = X.c ∧ y = X.b ∨ x = X.c ∧ y = X.d) X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	intros h	case right A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ ¬confluent RX	case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case right A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ ¬confluent RX TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	apply not_joins_a_d	case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ False	case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ joins RX X.a X.d	Please generate a tactic in lean4 to solve the state. STATE: case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ False TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	apply h	case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ joins RX X.a X.d	case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ wedge RX X.a X.d	Please generate a tactic in lean4 to solve the state. STATE: case right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ joins RX X.a X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	exists X.b	case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ wedge RX X.a X.d	case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a ∧ refl_trans_clos RX X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ wedge RX X.a X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a ∧ refl_trans_clos RX X.b X.d	case right.a.left A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a ∧ refl_trans_clos RX X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. constructor simp [RX] left; trivial constructor	case right.a.left A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d	case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. constructor simp [RX] right; trivial constructor . simp [RX]; right; trivial . constructor	case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a.left A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.left.b case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.a TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	simp [RX]	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.left.b case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a ∨ ?right.a.left.b = X.c case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.left.b case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	left	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a ∨ ?right.a.left.b = X.c case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.left.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a ∨ ?right.a.left.b = X.c case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case right.a.left.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.a X.a	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.left.b = X.a case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.left.b X.a case right.a.left.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.a X.a	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right.a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.a X.a TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.right.b case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.b X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	simp [RX]	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.right.b case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.a ∨ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.b ?right.a.right.b case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	right	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.a ∨ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.a ∨ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case right.a.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.c X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.b = X.c case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.b X.d case right.a.right.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.c X.d	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.c ?right.a.right.a.b case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.a.b X.d case right.a.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.c X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. simp [RX]; right; trivial	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.c ?right.a.right.a.b case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.a.b X.d case right.a.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.d X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.c ?right.a.right.a.b case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX ?right.a.right.a.b X.d case right.a.right.a.b A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ X TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	. constructor	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	simp [RX]	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.c ?right.a.right.a.b	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.b ∨ ?right.a.right.a.b = X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ RX X.c ?right.a.right.a.b TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	right	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.b ∨ ?right.a.right.a.b = X.d	case right.a.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.d	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.b ∨ ?right.a.right.a.b = X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	trivial	case right.a.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a.h A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ ?right.a.right.a.b = X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	weakly_confluent_does_not_imply_confluent	[532, 1]	[582, 20]	constructor	case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.d X.d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right.a.right.a.a A : Type R : A → A → Prop inhabited_A : Nonempty A h : confluent RX ⊢ refl_trans_clos RX X.d X.d TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros norm	A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ normalizing R → normalizing fun x x_1 => trans_clos R x x_1	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ normalizing fun x x_1 => trans_clos R x x_1	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ normalizing R → normalizing fun x x_1 => trans_clos R x x_1 TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply ind_normalizing	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ normalizing fun x x_1 => trans_clos R x x_1	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ ∀ (P : A → Prop), (∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x) → ∀ (x : A), P x	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ normalizing fun x x_1 => trans_clos R x x_1 TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros P ih	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ ∀ (P : A → Prop), (∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x) → ∀ (x : A), P x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x ⊢ ∀ (x : A), P x	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R ⊢ ∀ (P : A → Prop), (∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x) → ∀ (x : A), P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	let Q := λ x ↦ P x ∧ ∀ y, x ~>+ y → P y	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x ⊢ ∀ (x : A), P x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y ⊢ ∀ (x : A), P x	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x ⊢ ∀ (x : A), P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	have h : ∀ x, Q x := by intros x apply normalizing_ind R . intros x ih' constructor . apply ih intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial . intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial . trivial	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y ⊢ ∀ (x : A), P x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x ⊢ ∀ (x : A), P x	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y ⊢ ∀ (x : A), P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x ⊢ ∀ (x : A), P x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x x : A ⊢ P x	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x ⊢ ∀ (x : A), P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply (h x).1	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x x : A ⊢ P x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y h : ∀ (x : A), Q x x : A ⊢ P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros x	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y ⊢ ∀ (x : A), Q x	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ Q x	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y ⊢ ∀ (x : A), Q x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply normalizing_ind R	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ Q x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ ∀ (x : A), (∀ (y : A), R x y → Q y) → Q x case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ Q x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. intros x ih' constructor . apply ih intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial . intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ ∀ (x : A), (∀ (y : A), R x y → Q y) → Q x case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ ∀ (x : A), (∀ (y : A), R x y → Q y) → Q x case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. trivial	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros x ih'	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ ∀ (x : A), (∀ (y : A), R x y → Q y) → Q x	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ Q x	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ ∀ (x : A), (∀ (y : A), R x y → Q y) → Q x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	constructor	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ Q x	case a.left A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ P x case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ Q x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. apply ih intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial	case a.left A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ P x case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	Please generate a tactic in lean4 to solve the state. STATE: case a.left A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ P x case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. intros y red_x_y cases red_x_y . apply (ih' _ _).1; trivial . apply (ih' _ _).2 <;> trivial	case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply ih	case a.left A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ P x	case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	Please generate a tactic in lean4 to solve the state. STATE: case a.left A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ P x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros y red_x_y	case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	cases red_x_y	case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y	case a.left.a.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. apply (ih' _ _).1; trivial	case a.left.a.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. apply (ih' _ _).2 <;> trivial	case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply (ih' _ _).1	case a.left.a.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ R x y	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	trivial	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ R x y	no goals	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ R x y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply (ih' _ _).2 <;> trivial	case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.left.a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	intros y red_x_y	case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y	case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y ⊢ ∀ (y : A), trans_clos R x y → P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	cases red_x_y	case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y	case a.right.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.right A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A red_x_y : trans_clos R x y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. apply (ih' _ _).1; trivial	case a.right.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	Please generate a tactic in lean4 to solve the state. STATE: case a.right.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	. apply (ih' _ _).2 <;> trivial	case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply (ih' _ _).1	case a.right.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y	A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ R x y	Please generate a tactic in lean4 to solve the state. STATE: case a.right.base A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y : A a✝ : R x y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	apply (ih' _ _).2 <;> trivial	case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.right.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x✝ x : A ih' : ∀ (y : A), R x y → Q y y b✝ : A a✝¹ : R x b✝ a✝ : trans_clos R b✝ y ⊢ P y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	normalizes_trans	[585, 1]	[608, 26]	trivial	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm : normalizing R P : A → Prop ih : ∀ (x : A), (∀ (y : A), trans_clos R x y → P y) → P x Q : A → Prop := fun x => P x ∧ ∀ (y : A), trans_clos R x y → P y x : A ⊢ normalizing R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	intros conf'	A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ confluent' R → confluent R	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ confluent R	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ confluent' R → confluent R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	apply semi_confluent_implies_confluent	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ confluent R	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ semi_confluent R	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ confluent R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	intros x y z red_x_y red_x_z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ semi_confluent R	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R ⊢ semi_confluent R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	have h := refl_trans_is_trans_or_eq _ red_x_z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z ⊢ joins R y z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ∨ trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	cases h	case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ∨ trans_clos R x z ⊢ joins R y z	case a.inl A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : x = z ⊢ joins R y z case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ∨ trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	. case a.inl h => simp [← h] exists y; aesop	case a.inl A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : x = z ⊢ joins R y z case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z	case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a.inl A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : x = z ⊢ joins R y z case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	. apply conf' <;> aesop	case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	case a.inl h => simp [← h] exists y; aesop	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y z	no goals	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	simp [← h]	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y z	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y x	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	exists y	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y x	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ refl_trans_clos R y y ∧ refl_trans_clos R x y	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ joins R y x TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	aesop	A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ refl_trans_clos R y y ∧ refl_trans_clos R x y	no goals	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h : x = z ⊢ refl_trans_clos R y y ∧ refl_trans_clos R x y TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	confluent'_implies_confluent	[613, 1]	[623, 26]	apply conf' <;> aesop	case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.inr A : Type R : A → A → Prop inhabited_A : Nonempty A conf' : confluent' R x y z : A red_x_y : R x y red_x_z : refl_trans_clos R x z h✝ : trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	intros norm_R weak	A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ normalizing R → weakly_confluent R → confluent R	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R ⊢ confluent R	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A ⊢ normalizing R → weakly_confluent R → confluent R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	intros y z wedge	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R ⊢ confluent R	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z : A wedge : _root_.wedge R y z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R ⊢ confluent R TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	cases wedge	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z : A wedge : _root_.wedge R y z ⊢ joins R y z	case intro A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z w✝ : A h✝ : refl_trans_clos R w✝ y ∧ refl_trans_clos R w✝ z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z : A wedge : _root_.wedge R y z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	case intro x h => revert h z y have norm_trans := normalizes_trans _ norm_R have h := normalizes_ind (. ~>+ .) (λ x ↦ ∀ y z, x ~>* y ∧ x ~>* z → y ~>.<~ z) intros y z wedge (apply h <;> try trivial) <;> try apply norm_trans clear h intros x ih y z wedge cases wedge.1 . exists z; aesop . case a.step y' red_x_y' red_y'_y => cases wedge.2 . exists y; aesop . case step z' red_x_z' red_z'_z => have join1 := weak _ _ _ red_x_y' red_x_z' cases' join1 with w h' have join2 : y ~>.<~ w := by apply ih . apply trans_clos.base apply red_x_y' . aesop have join3 : z ~>.<~ w := by apply ih . apply trans_clos.base apply red_x_z' . aesop cases' join2 with w1 h1 cases' join3 with w2 h2 have red_x_w : x ~>+ w := by apply refl_trans_step_is_trans . apply red_x_z' . apply h'.2 have join4 : w1 ~>.<~ w2 := by apply ih . apply red_x_w . aesop cases' join4 with omega h3 exists omega; constructor . apply refl_trans_clos_transitive . apply h1.1 . apply h3.1 . apply refl_trans_clos_transitive . apply h2.1 . apply h3.2	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z x : A h : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	no goals	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z x : A h : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	revert h z y	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z x : A h : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R y z x : A h : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	have norm_trans := normalizes_trans _ norm_R	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	have h := normalizes_ind (. ~>+ .) (λ x ↦ ∀ y z, x ~>* y ∧ x ~>* z → y ~>.<~ z)	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	intros y z wedge	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z ⊢ ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	(apply h <;> try trivial) <;> try apply norm_trans	A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	Please generate a tactic in lean4 to solve the state. STATE: A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	clear h	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 h : (∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z) → ∀ (x : A), normalizes (fun x x_1 => trans_clos R x x_1) x → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	intros x ih y z wedge	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ ∀ (x : A), (∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z) → ∀ (y z : A), refl_trans_clos R x y ∧ refl_trans_clos R x z → joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	cases wedge.1	case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z	case a.refl A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z✝ : A wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z z : A wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z ⊢ joins R x z case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	. exists z; aesop	case a.refl A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z✝ : A wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z z : A wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z ⊢ joins R x z case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z	case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z	Please generate a tactic in lean4 to solve the state. STATE: case a.refl A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y z✝ : A wedge✝ : refl_trans_clos R x✝ y ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z z : A wedge : refl_trans_clos R x x ∧ refl_trans_clos R x z ⊢ joins R x z case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z TACTIC:
https://github.com/codyroux/traat-lean.git	f2babab84f81d4003446f476790022ac175d7236	Traat/chapter1.lean	newmans_lemma	[626, 1]	[685, 21]	. case a.step y' red_x_y' red_y'_y => cases wedge.2 . exists y; aesop . case step z' red_x_z' red_z'_z => have join1 := weak _ _ _ red_x_y' red_x_z' cases' join1 with w h' have join2 : y ~>.<~ w := by apply ih . apply trans_clos.base apply red_x_y' . aesop have join3 : z ~>.<~ w := by apply ih . apply trans_clos.base apply red_x_z' . aesop cases' join2 with w1 h1 cases' join3 with w2 h2 have red_x_w : x ~>+ w := by apply refl_trans_step_is_trans . apply red_x_z' . apply h'.2 have join4 : w1 ~>.<~ w2 := by apply ih . apply red_x_w . aesop cases' join4 with omega h3 exists omega; constructor . apply refl_trans_clos_transitive . apply h1.1 . apply h3.1 . apply refl_trans_clos_transitive . apply h2.1 . apply h3.2	case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.step A : Type R : A → A → Prop inhabited_A : Nonempty A norm_R : normalizing R weak : weakly_confluent R x✝ : A norm_trans : normalizing fun x x_1 => trans_clos R x x_1 y✝ z✝ : A wedge✝ : refl_trans_clos R x✝ y✝ ∧ refl_trans_clos R x✝ z✝ x : A ih : ∀ (y : A), trans_clos R x y → ∀ (y_1 z : A), refl_trans_clos R y y_1 ∧ refl_trans_clos R y z → joins R y_1 z y z : A wedge : refl_trans_clos R x y ∧ refl_trans_clos R x z b✝ : A a✝¹ : R x b✝ a✝ : refl_trans_clos R b✝ y ⊢ joins R y z TACTIC:

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