internlm/Lean-Github · Datasets at Hugging Face

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
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	cases x1_mem	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 ⊢ x1 = x2	case inl α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 h✝ : x1 = xs_hd ⊢ x1 = x2 case inr α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 h✝ : x1 ∈ xs_tl ⊢ x1 = x2
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	case inl x1_mem_left => contradiction	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_left : x1 = xs_hd ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	case inr x1_mem_right => cases x2_mem case inl x2_mem_left => contradiction case inr x2_mem_right => simp only [List.InjOn] at a1_left apply a1_left x1_mem_right x2_mem_right a2	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	contradiction	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_left : x1 = xs_hd ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	cases x2_mem	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl ⊢ x1 = x2	case inl α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl h✝ : x2 = xs_hd ⊢ x1 = x2 case inr α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl h✝ : x2 ∈ xs_tl ⊢ x1 = x2
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	case inl x2_mem_left => contradiction	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_left : x2 = xs_hd ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	case inr x2_mem_right => simp only [List.InjOn] at a1_left apply a1_left x1_mem_right x2_mem_right a2	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	contradiction	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_left : x2 = xs_hd ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	simp only [List.InjOn] at a1_left	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : ∀ ⦃x₁ : α⦄, x₁ ∈ xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_tl → f x₁ = f x₂ → x₁ = x₂ a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	apply a1_left x1_mem_right x2_mem_right a2	α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : ∀ ⦃x₁ : α⦄, x₁ ∈ xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_tl → f x₁ = f x₂ → x₁ = x₂ a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	constructor	case right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl	case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	simp only [Function.updateITE]	case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd	case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ (if True then ys_hd else f xs_hd) = ys_hd
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	simp	case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ (if True then ys_hd else f xs_hd) = ys_hd	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	simp only [← a1_right]	case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl	case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = map f xs_tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	apply Function.updateITE_not_mem_list	case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = map f xs_tl	case right.right.h1 α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ xs_hd ∉ xs_tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	List.nodup_eq_len_imp_exists_bijon	[162, 1]	[256, 30]	exact h2_left	case right.right.h1 α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ xs_hd ∉ xs_tl	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/List.lean	nodup_eq_len_imp_eqv	[259, 1]	[267, 11]	sorry	α : Type inst✝ : DecidableEq α xs ys : List α h1 : xs.length = ys.length h2 : xs.Nodup h3 : ys.Nodup ⊢ ∃ f, List.map (⇑f) xs = ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	DFA.mem_accepts	[91, 1]	[99, 74]	rfl	α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ D : DFA α σ input : List α ⊢ D.accepts input ↔ D.eval_from D.starting_state input ∈ D.accepting_state_list	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	NFA_to_DFA_is_equiv	[105, 1]	[117, 8]	ext cs	α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ ⊢ N.to_DFA.accepts = N.accepts	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.accepts cs ↔ N.accepts cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	NFA_to_DFA_is_equiv	[105, 1]	[117, 8]	simp only [DFA.mem_accepts]	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.accepts cs ↔ N.accepts cs	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ N.accepts cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	NFA_to_DFA_is_equiv	[105, 1]	[117, 8]	simp only [NFA.mem_accepts]	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ N.accepts cs	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	NFA_to_DFA_is_equiv	[105, 1]	[117, 8]	simp only [NFA.to_DFA]	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ { step := N.eval_one, starting_state := N.starting_state_list, accepting_state_list := sorryAx (List (List σ)) }.eval_from N.starting_state_list cs ∈ sorryAx (List (List σ)) ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFA.lean	NFA_to_DFA_is_equiv	[105, 1]	[117, 8]	sorry	case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ { step := N.eval_one, starting_state := N.starting_state_list, accepting_state_list := sorryAx (List (List σ)) }.eval_from N.starting_state_list cs ∈ sorryAx (List (List σ)) ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	induction F generalizing vs	F : Formula vs : List VarName v : VarName Γ : List (VarName × MetaVarName) h1 : NoMetaVarAndAllFreeInList vs F h2 : v ∉ vs ⊢ NotFree Γ v F	case meta_var_ v : VarName Γ : List (VarName × MetaVarName) a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ a✝) case pred_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ a✝¹ a✝) case eq_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (eq_ a✝¹ a✝) case true_ v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_ case not_ v : VarName Γ : List (VarName × MetaVarName) a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ h2 : v ∉ vs ⊢ NotFree Γ v a✝.not_ case imp_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ a✝ : Formula a_ih✝¹ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ → v ∉ vs → NotFree Γ v a✝¹ a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (a✝¹.imp_ a✝) case forall_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (forall_ a✝¹ a✝) case def_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction	v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ X)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	case pred_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	case true_ => unfold NotFree simp	v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	case not_ phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree exact phi_ih vs h1 h2	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v ∉ vs ⊢ NotFree Γ v phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	case def_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ X)	v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h2 : v ∉ vs h1 : False ⊢ NotFree Γ v (meta_var_ X)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	contradiction	v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h2 : v ∉ vs h1 : False ⊢ NotFree Γ v (meta_var_ X)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ X xs)	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (pred_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (pred_ X xs)	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	tauto	v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) h2 : v ∉ vs ⊢ NotFree Γ v (eq_ x y)	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ NotFree Γ v (eq_ x y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ NotFree Γ v (eq_ x y)	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ ¬x = v ∧ ¬y = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	cases h1	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ ¬x = v ∧ ¬y = v	case intro v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs left✝ : x ∈ vs right✝ : y ∈ vs ⊢ ¬x = v ∧ ¬y = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	constructor	v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v ∧ ¬y = v	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬y = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	intro contra	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	apply h2	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ False	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	subst contra	case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ v ∈ vs	case left Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x ∉ vs ⊢ x ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact h1_left	case left Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x ∉ vs ⊢ x ∈ vs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	intro contra	case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬y = v	case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	apply h2	case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ False	case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	subst contra	case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ v ∈ vs	case right Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y ∉ vs ⊢ y ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact h1_right	case right Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y ∉ vs ⊢ y ∈ vs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_	v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ True
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	simp	v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ True	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v ∉ vs ⊢ NotFree Γ v phi.not_	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi.not_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi.not_	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact phi_ih vs h1 h2	v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) h2 : v ∉ vs ⊢ NotFree Γ v (phi.imp_ psi)	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v (phi.imp_ psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v (phi.imp_ psi)	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	cases h1	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi	case intro v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs left✝ : NoMetaVarAndAllFreeInList vs phi right✝ : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	constructor	v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi	case left v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi case right v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact phi_ih vs h1_left h2	case left v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact psi_ih vs h1_right h2	case right v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) h2 : v ∉ vs ⊢ NotFree Γ v (forall_ x phi)	v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ NotFree Γ v (forall_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ NotFree Γ v (forall_ x phi)	v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ x = v ∨ NotFree Γ v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	by_cases c1 : x = v	v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ x = v ∨ NotFree Γ v phi	case pos v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v ∨ NotFree Γ v phi case neg v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ x = v ∨ NotFree Γ v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	left	case pos v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v ∨ NotFree Γ v phi	case pos.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact c1	case pos.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	right	case neg v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ x = v ∨ NotFree Γ v phi	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ NotFree Γ v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	apply phi_ih (x :: vs) h1	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ NotFree Γ v phi	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ x :: vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	simp	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ x :: vs	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ ¬v = x ∧ v ∉ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	push_neg	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ ¬v = x ∧ v ∉ vs	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x ∧ v ∉ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	constructor	case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x ∧ v ∉ vs	case neg.h.left v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x case neg.h.right v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	tauto	case neg.h.left v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	exact h2	case neg.h.right v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ vs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NoMetaVarAndAllFreeInList at h1	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (def_ X xs)	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	unfold NotFree	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (def_ X xs)	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.all_free_in_list_and_not_in_list_imp_not_free	[572, 1]	[641, 10]	tauto	v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	induction F generalizing vs	F : Formula vs : List VarName h1 : NoMetaVarAndAllFreeInList vs F ⊢ F.metaVarSet = ∅	case meta_var_ a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) ⊢ (meta_var_ a✝).metaVarSet = ∅ case pred_ a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) ⊢ (pred_ a✝¹ a✝).metaVarSet = ∅ case eq_ a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) ⊢ (eq_ a✝¹ a✝).metaVarSet = ∅ case true_ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅ case not_ a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ ⊢ a✝.not_.metaVarSet = ∅ case imp_ a✝¹ a✝ : Formula a_ih✝¹ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ → a✝¹.metaVarSet = ∅ a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) ⊢ (a✝¹.imp_ a✝).metaVarSet = ∅ case forall_ a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) ⊢ (forall_ a✝¹ a✝).metaVarSet = ∅ case def_ a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) ⊢ (def_ a✝¹ a✝).metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction	X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) ⊢ (meta_var_ X).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case pred_ X xs => rfl	X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) ⊢ (pred_ X xs).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case eq_ x y => rfl	x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) ⊢ (eq_ x y).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case true_ => rfl	vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case not_ phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet exact phi_ih vs h1	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ ⊢ phi.not_.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case imp_ phi psi phi_ih psi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet cases h1 case intro h1_left h1_right => simp only [phi_ih vs h1_left] simp only [psi_ih vs h1_right] simp	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) ⊢ (phi.imp_ psi).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case forall_ x phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet exact phi_ih (x :: vs) h1	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) ⊢ (forall_ x phi).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case def_ X xs => rfl	X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) ⊢ (def_ X xs).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold NoMetaVarAndAllFreeInList at h1	X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) ⊢ (meta_var_ X).metaVarSet = ∅	X : MetaVarName vs : List VarName h1 : False ⊢ (meta_var_ X).metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	contradiction	X : MetaVarName vs : List VarName h1 : False ⊢ (meta_var_ X).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	rfl	X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) ⊢ (pred_ X xs).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	rfl	x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) ⊢ (eq_ x y).metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	rfl	vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold NoMetaVarAndAllFreeInList at h1	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ ⊢ phi.not_.metaVarSet = ∅	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.not_.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold Formula.metaVarSet	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.not_.metaVarSet = ∅	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	exact phi_ih vs h1	phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold NoMetaVarAndAllFreeInList at h1	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) ⊢ (phi.imp_ psi).metaVarSet = ∅	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ (phi.imp_ psi).metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold Formula.metaVarSet	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ (phi.imp_ psi).metaVarSet = ∅	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	cases h1	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅	case intro phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName left✝ : NoMetaVarAndAllFreeInList vs phi right✝ : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	case intro h1_left h1_right => simp only [phi_ih vs h1_left] simp only [psi_ih vs h1_right] simp	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	simp only [phi_ih vs h1_left]	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ psi.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	simp only [psi_ih vs h1_right]	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ psi.metaVarSet = ∅	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ ∅ = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	simp	phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ ∅ = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold NoMetaVarAndAllFreeInList at h1	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) ⊢ (forall_ x phi).metaVarSet = ∅	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ (forall_ x phi).metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	unfold Formula.metaVarSet	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ (forall_ x phi).metaVarSet = ∅	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ phi.metaVarSet = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	exact phi_ih (x :: vs) h1	x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ phi.metaVarSet = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/MM0/MM0.lean	MM0.no_meta_var_imp_meta_var_set_is_empty	[644, 1]	[681, 8]	rfl	X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) ⊢ (def_ X xs).metaVarSet = ∅	no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

cases x1_mem

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 ⊢ x1 = x2

case inl α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 h✝ : x1 = xs_hd ⊢ x1 = x2 case inr α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 h✝ : x1 ∈ xs_tl ⊢ x1 = x2

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

case inl x1_mem_left => contradiction

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_left : x1 = xs_hd ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

case inr x1_mem_right => cases x2_mem case inl x2_mem_left => contradiction case inr x2_mem_right => simp only [List.InjOn] at a1_left apply a1_left x1_mem_right x2_mem_right a2

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

contradiction

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_left : x1 = xs_hd ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

cases x2_mem

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl ⊢ x1 = x2

case inl α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl h✝ : x2 = xs_hd ⊢ x1 = x2 case inr α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl h✝ : x2 ∈ xs_tl ⊢ x1 = x2

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

case inl x2_mem_left => contradiction

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_left : x2 = xs_hd ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

case inr x2_mem_right => simp only [List.InjOn] at a1_left apply a1_left x1_mem_right x2_mem_right a2

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

contradiction

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_left : x2 = xs_hd ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

simp only [List.InjOn] at a1_left

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : ∀ ⦃x₁ : α⦄, x₁ ∈ xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_tl → f x₁ = f x₂ → x₁ = x₂ a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

apply a1_left x1_mem_right x2_mem_right a2

α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : ∀ ⦃x₁ : α⦄, x₁ ∈ xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_tl → f x₁ = f x₂ → x₁ = x₂ a1_right : map f xs_tl = ys_tl x1 x2 : α c1 : ¬x1 = xs_hd c2 : ¬x2 = xs_hd a2 : f x1 = f x2 x1_mem_right : x1 ∈ xs_tl x2_mem_right : x2 ∈ xs_tl ⊢ x1 = x2

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

constructor

case right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl

case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

simp only [Function.updateITE]

case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd

case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ (if True then ys_hd else f xs_hd) = ys_hd

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

simp

case right.left α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ (if True then ys_hd else f xs_hd) = ys_hd

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

simp only [← a1_right]

case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl

case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = map f xs_tl

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

apply Function.updateITE_not_mem_list

case right.right α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ map (Function.updateITE f xs_hd ys_hd) xs_tl = map f xs_tl

case right.right.h1 α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ xs_hd ∉ xs_tl

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

List.nodup_eq_len_imp_exists_bijon

[162, 1]

[256, 30]

exact h2_left

case right.right.h1 α : Type inst✝ : DecidableEq α xs_hd : α xs_tl : List α ys_hd : α ys_tl : List α h1 : xs_tl.length = ys_tl.length h2_left : xs_hd ∉ xs_tl h2_right : xs_tl.Nodup h3_left : ys_hd ∉ ys_tl h3_right : ys_tl.Nodup f : α → α a1_left : InjOn f xs_tl a1_right : map f xs_tl = ys_tl ⊢ xs_hd ∉ xs_tl

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/List.lean

nodup_eq_len_imp_eqv

[259, 1]

[267, 11]

sorry

α : Type inst✝ : DecidableEq α xs ys : List α h1 : xs.length = ys.length h2 : xs.Nodup h3 : ys.Nodup ⊢ ∃ f, List.map (⇑f) xs = ys

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

DFA.mem_accepts

[91, 1]

[99, 74]

rfl

α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ D : DFA α σ input : List α ⊢ D.accepts input ↔ D.eval_from D.starting_state input ∈ D.accepting_state_list

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

NFA_to_DFA_is_equiv

[105, 1]

[117, 8]

ext cs

α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ ⊢ N.to_DFA.accepts = N.accepts

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.accepts cs ↔ N.accepts cs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

NFA_to_DFA_is_equiv

[105, 1]

[117, 8]

simp only [DFA.mem_accepts]

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.accepts cs ↔ N.accepts cs

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ N.accepts cs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

NFA_to_DFA_is_equiv

[105, 1]

[117, 8]

simp only [NFA.mem_accepts]

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ N.accepts cs

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

NFA_to_DFA_is_equiv

[105, 1]

[117, 8]

simp only [NFA.to_DFA]

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ N.to_DFA.eval_from N.to_DFA.starting_state cs ∈ N.to_DFA.accepting_state_list ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ { step := N.eval_one, starting_state := N.starting_state_list, accepting_state_list := sorryAx (List (List σ)) }.eval_from N.starting_state_list cs ∈ sorryAx (List (List σ)) ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFA.lean

NFA_to_DFA_is_equiv

[105, 1]

[117, 8]

sorry

case h.a α : Type inst✝¹ : DecidableEq α σ : Type inst✝ : DecidableEq σ N : NFA α σ cs : List α ⊢ { step := N.eval_one, starting_state := N.starting_state_list, accepting_state_list := sorryAx (List (List σ)) }.eval_from N.starting_state_list cs ∈ sorryAx (List (List σ)) ↔ ∃ s ∈ N.eval_from N.starting_state_list cs, s ∈ N.accepting_state_list

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

induction F generalizing vs

F : Formula vs : List VarName v : VarName Γ : List (VarName × MetaVarName) h1 : NoMetaVarAndAllFreeInList vs F h2 : v ∉ vs ⊢ NotFree Γ v F

case meta_var_ v : VarName Γ : List (VarName × MetaVarName) a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ a✝) case pred_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ a✝¹ a✝) case eq_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (eq_ a✝¹ a✝) case true_ v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_ case not_ v : VarName Γ : List (VarName × MetaVarName) a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ h2 : v ∉ vs ⊢ NotFree Γ v a✝.not_ case imp_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ a✝ : Formula a_ih✝¹ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ → v ∉ vs → NotFree Γ v a✝¹ a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (a✝¹.imp_ a✝) case forall_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → v ∉ vs → NotFree Γ v a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (forall_ a✝¹ a✝) case def_ v : VarName Γ : List (VarName × MetaVarName) a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) h2 : v ∉ vs ⊢ NotFree Γ v (def_ a✝¹ a✝)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction

v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ X)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

case pred_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ X xs)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

case true_ => unfold NotFree simp

v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

case not_ phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree exact phi_ih vs h1 h2

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v ∉ vs ⊢ NotFree Γ v phi.not_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

case def_ X xs => unfold NoMetaVarAndAllFreeInList at h1 unfold NotFree tauto

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (def_ X xs)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) h2 : v ∉ vs ⊢ NotFree Γ v (meta_var_ X)

v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h2 : v ∉ vs h1 : False ⊢ NotFree Γ v (meta_var_ X)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

contradiction

v : VarName Γ : List (VarName × MetaVarName) X : MetaVarName vs : List VarName h2 : v ∉ vs h1 : False ⊢ NotFree Γ v (meta_var_ X)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (pred_ X xs)

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (pred_ X xs)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (pred_ X xs)

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

tauto

v : VarName Γ : List (VarName × MetaVarName) X : PredName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) h2 : v ∉ vs ⊢ NotFree Γ v (eq_ x y)

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ NotFree Γ v (eq_ x y)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ NotFree Γ v (eq_ x y)

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ ¬x = v ∧ ¬y = v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

cases h1

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1 : x ∈ vs ∧ y ∈ vs ⊢ ¬x = v ∧ ¬y = v

case intro v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs left✝ : x ∈ vs right✝ : y ∈ vs ⊢ ¬x = v ∧ ¬y = v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

constructor

v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v ∧ ¬y = v

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬y = v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

intro contra

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬x = v

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ False

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

apply h2

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ False

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ v ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

subst contra

case left v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : x = v ⊢ v ∈ vs

case left Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x ∉ vs ⊢ x ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact h1_left

case left Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : x ∉ vs ⊢ x ∈ vs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

intro contra

case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs ⊢ ¬y = v

case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ False

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

apply h2

case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ False

case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ v ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

subst contra

case right v : VarName Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h2 : v ∉ vs h1_left : x ∈ vs h1_right : y ∈ vs contra : y = v ⊢ v ∈ vs

case right Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y ∉ vs ⊢ y ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact h1_right

case right Γ : List (VarName × MetaVarName) x y : VarName vs : List VarName h1_left : x ∈ vs h1_right : y ∈ vs h2 : y ∉ vs ⊢ y ∈ vs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ NotFree Γ v true_

v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ True

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

simp

v : VarName Γ : List (VarName × MetaVarName) vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ h2 : v ∉ vs ⊢ True

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ h2 : v ∉ vs ⊢ NotFree Γ v phi.not_

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi.not_

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi.not_

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact phi_ih vs h1 h2

v : VarName Γ : List (VarName × MetaVarName) phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ⊢ NotFree Γ v phi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) h2 : v ∉ vs ⊢ NotFree Γ v (phi.imp_ psi)

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v (phi.imp_ psi)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v (phi.imp_ psi)

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

cases h1

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi

case intro v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs left✝ : NoMetaVarAndAllFreeInList vs phi right✝ : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

constructor

v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi ∧ NotFree Γ v psi

case left v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi case right v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact phi_ih vs h1_left h2

case left v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v phi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact psi_ih vs h1_right h2

case right v : VarName Γ : List (VarName × MetaVarName) phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → v ∉ vs → NotFree Γ v psi vs : List VarName h2 : v ∉ vs h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ NotFree Γ v psi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) h2 : v ∉ vs ⊢ NotFree Γ v (forall_ x phi)

v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ NotFree Γ v (forall_ x phi)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ NotFree Γ v (forall_ x phi)

v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ x = v ∨ NotFree Γ v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

by_cases c1 : x = v

v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ x = v ∨ NotFree Γ v phi

case pos v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v ∨ NotFree Γ v phi case neg v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ x = v ∨ NotFree Γ v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

left

case pos v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v ∨ NotFree Γ v phi

case pos.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact c1

case pos.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : x = v ⊢ x = v

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

right

case neg v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ x = v ∨ NotFree Γ v phi

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ NotFree Γ v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

apply phi_ih (x :: vs) h1

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ NotFree Γ v phi

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ x :: vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

simp

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ x :: vs

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ ¬v = x ∧ v ∉ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

push_neg

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ ¬v = x ∧ v ∉ vs

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x ∧ v ∉ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

constructor

case neg.h v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x ∧ v ∉ vs

case neg.h.left v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x case neg.h.right v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

tauto

case neg.h.left v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ≠ x

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

exact h2

case neg.h.right v : VarName Γ : List (VarName × MetaVarName) x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → v ∉ vs → NotFree Γ v phi vs : List VarName h2 : v ∉ vs h1 : NoMetaVarAndAllFreeInList (x :: vs) phi c1 : ¬x = v ⊢ v ∉ vs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NoMetaVarAndAllFreeInList at h1

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) h2 : v ∉ vs ⊢ NotFree Γ v (def_ X xs)

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (def_ X xs)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

unfold NotFree

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ NotFree Γ v (def_ X xs)

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.all_free_in_list_and_not_in_list_imp_not_free

[572, 1]

[641, 10]

tauto

v : VarName Γ : List (VarName × MetaVarName) X : DefName xs vs : List VarName h2 : v ∉ vs h1 : xs ⊆ vs ⊢ v ∉ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

induction F generalizing vs

F : Formula vs : List VarName h1 : NoMetaVarAndAllFreeInList vs F ⊢ F.metaVarSet = ∅

case meta_var_ a✝ : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ a✝) ⊢ (meta_var_ a✝).metaVarSet = ∅ case pred_ a✝¹ : PredName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ a✝¹ a✝) ⊢ (pred_ a✝¹ a✝).metaVarSet = ∅ case eq_ a✝¹ a✝ : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ a✝¹ a✝) ⊢ (eq_ a✝¹ a✝).metaVarSet = ∅ case true_ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅ case not_ a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs a✝.not_ ⊢ a✝.not_.metaVarSet = ∅ case imp_ a✝¹ a✝ : Formula a_ih✝¹ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝¹ → a✝¹.metaVarSet = ∅ a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (a✝¹.imp_ a✝) ⊢ (a✝¹.imp_ a✝).metaVarSet = ∅ case forall_ a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs a✝ → a✝.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ a✝¹ a✝) ⊢ (forall_ a✝¹ a✝).metaVarSet = ∅ case def_ a✝¹ : DefName a✝ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ a✝¹ a✝) ⊢ (def_ a✝¹ a✝).metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case meta_var_ X => unfold NoMetaVarAndAllFreeInList at h1 contradiction

X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) ⊢ (meta_var_ X).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case pred_ X xs => rfl

X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) ⊢ (pred_ X xs).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case eq_ x y => rfl

x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) ⊢ (eq_ x y).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case true_ => rfl

vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case not_ phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet exact phi_ih vs h1

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ ⊢ phi.not_.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case imp_ phi psi phi_ih psi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet cases h1 case intro h1_left h1_right => simp only [phi_ih vs h1_left] simp only [psi_ih vs h1_right] simp

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) ⊢ (phi.imp_ psi).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case forall_ x phi phi_ih => unfold NoMetaVarAndAllFreeInList at h1 unfold Formula.metaVarSet exact phi_ih (x :: vs) h1

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) ⊢ (forall_ x phi).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case def_ X xs => rfl

X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) ⊢ (def_ X xs).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold NoMetaVarAndAllFreeInList at h1

X : MetaVarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (meta_var_ X) ⊢ (meta_var_ X).metaVarSet = ∅

X : MetaVarName vs : List VarName h1 : False ⊢ (meta_var_ X).metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

contradiction

X : MetaVarName vs : List VarName h1 : False ⊢ (meta_var_ X).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

rfl

X : PredName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (pred_ X xs) ⊢ (pred_ X xs).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

rfl

x y : VarName vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (eq_ x y) ⊢ (eq_ x y).metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

rfl

vs : List VarName h1 : NoMetaVarAndAllFreeInList vs true_ ⊢ true_.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold NoMetaVarAndAllFreeInList at h1

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi.not_ ⊢ phi.not_.metaVarSet = ∅

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.not_.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold Formula.metaVarSet

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.not_.metaVarSet = ∅

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

exact phi_ih vs h1

phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ⊢ phi.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold NoMetaVarAndAllFreeInList at h1

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (phi.imp_ psi) ⊢ (phi.imp_ psi).metaVarSet = ∅

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ (phi.imp_ psi).metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold Formula.metaVarSet

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ (phi.imp_ psi).metaVarSet = ∅

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

cases h1

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs phi ∧ NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅

case intro phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName left✝ : NoMetaVarAndAllFreeInList vs phi right✝ : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

case intro h1_left h1_right => simp only [phi_ih vs h1_left] simp only [psi_ih vs h1_right] simp

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

simp only [phi_ih vs h1_left]

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ phi.metaVarSet ∪ psi.metaVarSet = ∅

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ psi.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

simp only [psi_ih vs h1_right]

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ psi.metaVarSet = ∅

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ ∅ = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

simp

phi psi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ psi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs psi → psi.metaVarSet = ∅ vs : List VarName h1_left : NoMetaVarAndAllFreeInList vs phi h1_right : NoMetaVarAndAllFreeInList vs psi ⊢ ∅ ∪ ∅ = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold NoMetaVarAndAllFreeInList at h1

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (forall_ x phi) ⊢ (forall_ x phi).metaVarSet = ∅

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ (forall_ x phi).metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

unfold Formula.metaVarSet

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ (forall_ x phi).metaVarSet = ∅

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ phi.metaVarSet = ∅

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

exact phi_ih (x :: vs) h1

x : VarName phi : Formula phi_ih : ∀ (vs : List VarName), NoMetaVarAndAllFreeInList vs phi → phi.metaVarSet = ∅ vs : List VarName h1 : NoMetaVarAndAllFreeInList (x :: vs) phi ⊢ phi.metaVarSet = ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/MM0/MM0.lean

MM0.no_meta_var_imp_meta_var_set_is_empty

[644, 1]

[681, 8]

rfl

X : DefName xs vs : List VarName h1 : NoMetaVarAndAllFreeInList vs (def_ X xs) ⊢ (def_ X xs).metaVarSet = ∅

no goals