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/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	apply reachable_from_list_mono g xs (xs ++ ys)	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g (xs ++ ys)	case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ xs ⊆ xs ++ ys case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	simp	case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ xs ⊆ xs ++ ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	exact left	case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	apply reachable_from_list_mono g ys (xs ++ ys)	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)	case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ ys ⊆ xs ++ ys case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	simp	case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ ys ⊆ xs ++ ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	exact right	case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append	[695, 1]	[704, 48]	apply Set.eq_of_subset_of_subset	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) = reachable_from_list g xs ∪ reachable_from_list g ys	case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) ⊆ reachable_from_list g xs ∪ reachable_from_list g ys case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g xs ∪ reachable_from_list g ys ⊆ reachable_from_list g (xs ++ ys)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append	[695, 1]	[704, 48]	apply reachable_from_list_of_append_left	case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) ⊆ reachable_from_list g xs ∪ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append	[695, 1]	[704, 48]	apply reachable_from_list_of_append_right	case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g xs ∪ reachable_from_list g ys ⊆ reachable_from_list g (xs ++ ys)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_cons	[707, 1]	[716, 9]	simp only [← reachable_from_list_of_append]	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g [x] ∪ reachable_from_list g ys	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g ([x] ++ ys)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_cons	[707, 1]	[716, 9]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g ([x] ++ ys)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	simp only [Set.subset_def]	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x]	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	intro a a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x]	Node : Type inst✝ : DecidableEq Node g : Graph Node x a : Node a1 : a ∈ reachable_from_list g (direct_succ_list g x) ⊢ a ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	induction a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x a : Node a1 : a ∈ reachable_from_list g (direct_succ_list g x) ⊢ a ∈ reachable_from_list g [x]	case base Node : Type inst✝ : DecidableEq Node g : Graph Node x a x✝ : Node a✝ : x✝ ∈ direct_succ_list g x ⊢ x✝ ∈ reachable_from_list g [x] case step Node : Type inst✝ : DecidableEq Node g : Graph Node x a x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g (direct_succ_list g x) e✝.1 a_ih✝ : e✝.1 ∈ reachable_from_list g [x] ⊢ x✝ ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	case _ x y ih => obtain s1 := mem_direct_succ_list_iff g x y simp only [s1] at ih apply Exists.elim ih intro ys a2 cases a2 case _ a2_left a2_right => apply reachable_from_list.step y (x, ys) a2_right a2_left apply reachable_from_list.base x simp	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x ⊢ y ∈ reachable_from_list g [x]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	case _ x e ih_1 ih_2 _ ih_4 => exact reachable_from_list.step x e ih_1 ih_2 ih_4	Node : Type inst✝ : DecidableEq Node g : Graph Node x✝ a x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g (direct_succ_list g x✝) e.1 ih_4 : e.1 ∈ reachable_from_list g [x✝] ⊢ x ∈ reachable_from_list g [x✝]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	obtain s1 := mem_direct_succ_list_iff g x y	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x ⊢ y ∈ reachable_from_list g [x]	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	simp only [s1] at ih	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	apply Exists.elim ih	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ ∀ (a : List Node), y ∈ a ∧ (x, a) ∈ g → y ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	intro ys a2	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ ∀ (a : List Node), y ∈ a ∧ (x, a) ∈ g → y ∈ reachable_from_list g [x]	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2 : y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	cases a2	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2 : y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]	case intro Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node left✝ : y ∈ ys right✝ : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	case _ a2_left a2_right => apply reachable_from_list.step y (x, ys) a2_right a2_left apply reachable_from_list.base x simp	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	apply reachable_from_list.step y (x, ys) a2_right a2_left	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ reachable_from_list g [x] (x, ys).1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	apply reachable_from_list.base x	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ reachable_from_list g [x] (x, ys).1	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ x ∈ [x]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	simp	Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ x ∈ [x]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list	[719, 1]	[740, 56]	exact reachable_from_list.step x e ih_1 ih_2 ih_4	Node : Type inst✝ : DecidableEq Node g : Graph Node x✝ a x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g (direct_succ_list g x✝) e.1 ih_4 : e.1 ∈ reachable_from_list g [x✝] ⊢ x ∈ reachable_from_list g [x✝]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp only [Set.subset_def]	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ reachable_from_list g (list_direct_succ_list g xs) ⊆ reachable_from_list g xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ reachable_from_list g (list_direct_succ_list g xs), x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	intro x a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ reachable_from_list g (list_direct_succ_list g xs), x ∈ reachable_from_list g xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g xs) ⊢ x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	induction xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g xs) ⊢ x ∈ reachable_from_list g xs	case nil Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g [] case cons Node : Type inst✝ : DecidableEq Node g : Graph Node x head✝ : Node tail✝ : List Node tail_ih✝ : x ∈ reachable_from_list g (list_direct_succ_list g tail✝) → x ∈ reachable_from_list g tail✝ a1 : x ∈ reachable_from_list g (list_direct_succ_list g (head✝ :: tail✝)) ⊢ x ∈ reachable_from_list g (head✝ :: tail✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	case nil => simp only [list_direct_succ_list] at a1 simp at a1 exact a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	case cons hd tl ih => simp only [list_direct_succ_list_cons] at a1 simp only [reachable_from_list_of_cons g hd tl] simp simp only [reachable_from_list_of_append] at a1 simp at a1 cases a1 case _ left => left obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd exact Set.mem_of_subset_of_mem s1 left case _ right => right exact ih right	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (list_direct_succ_list g (hd :: tl)) ⊢ x ∈ reachable_from_list g (hd :: tl)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp only [list_direct_succ_list] at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g []	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (List.map (fun x => direct_succ_list g x) []).join ⊢ x ∈ reachable_from_list g []
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (List.map (fun x => direct_succ_list g x) []).join ⊢ x ∈ reachable_from_list g []	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g [] ⊢ x ∈ reachable_from_list g []
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	exact a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g [] ⊢ x ∈ reachable_from_list g []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp only [list_direct_succ_list_cons] at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (list_direct_succ_list g (hd :: tl)) ⊢ x ∈ reachable_from_list g (hd :: tl)	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g (hd :: tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp only [reachable_from_list_of_cons g hd tl]	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g (hd :: tl)	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∪ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∪ reachable_from_list g tl	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp only [reachable_from_list_of_append] at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∪ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	simp at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∪ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∨ x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	cases a1	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∨ x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	case inl Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl h✝ : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl case inr Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl h✝ : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	case _ left => left obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd exact Set.mem_of_subset_of_mem s1 left	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	case _ right => right exact ih right	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	left	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd]	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) s1 : reachable_from_list g (direct_succ_list g hd) ⊆ reachable_from_list g [hd] ⊢ x ∈ reachable_from_list g [hd]
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	exact Set.mem_of_subset_of_mem s1 left	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) s1 : reachable_from_list g (direct_succ_list g hd) ⊆ reachable_from_list g [hd] ⊢ x ∈ reachable_from_list g [hd]	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	right	Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list	[743, 1]	[771, 23]	exact ih right	case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g tl	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	induction stack, visited using dft_aux.induct g	Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset	case case1 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node ⊢ ↑(dft_aux g [] visited✝).toFinset ⊆ reachable_from_list g [] ∪ ↑visited✝.toFinset case case2 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∈ visited✝ ih1✝ : ↑(dft_aux g xs✝ visited✝).toFinset ⊆ reachable_from_list g xs✝ ∪ ↑visited✝.toFinset ⊢ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset ⊆ reachable_from_list g (x✝ :: xs✝) ∪ ↑visited✝.toFinset case case3 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∉ visited✝ ih1✝ : ↑(dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)).toFinset ⊆ reachable_from_list g (direct_succ_list g x✝ ++ xs✝) ∪ ↑(x✝ :: visited✝).toFinset ⊢ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset ⊆ reachable_from_list g (x✝ :: xs✝) ∪ ↑visited✝.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	case _ visited => simp only [dft_aux] simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑(dft_aux g [] visited).toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	case _ visited x stack c1 ih => simp only [reachable_from_list_of_cons g x stack] simp only [dft_aux] simp only [if_pos c1] simp only [Set.union_assoc ] exact Set.subset_union_of_subset_right ih (reachable_from_list g [x])	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑(dft_aux g [] visited).toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑visited.toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑visited.toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [reachable_from_list_of_cons g x stack]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [if_pos c1]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [Set.union_assoc ]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ (reachable_from_list g stack ∪ ↑visited.toFinset)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	exact Set.subset_union_of_subset_right ih (reachable_from_list g [x])	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ (reachable_from_list g stack ∪ ↑visited.toFinset)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [if_neg c1]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [reachable_from_list_of_append g (direct_succ_list g x) stack] at ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [reachable_from_list_of_cons g x stack]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	trans (reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ (x :: visited).toFinset.toSet)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	exact ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	have s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g x	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp only [Set.subset_def] at *	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ∀ x_1 ∈ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset, x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset, x_1 ∈ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	simp at *	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ∀ x_1 ∈ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset, x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset, x_1 ∈ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ih : ∀ x_1 ∈ dft_aux g (direct_succ_list g x ++ stack) (x :: visited), x_1 = x ∨ (x_1 ∈ reachable_from_list g (direct_succ_list g x) ∨ x_1 ∈ reachable_from_list g stack) ∨ x_1 ∈ visited ⊢ ((x ∈ reachable_from_list g [x] ∨ x ∈ reachable_from_list g stack) ∨ x ∈ visited) ∧ ∀ (a : Node), (a ∈ reachable_from_list g (direct_succ_list g x) ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited → (a ∈ reachable_from_list g [x] ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_is_subset_of_reachable_from_list_and_visited	[774, 1]	[809, 14]	tauto	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ih : ∀ x_1 ∈ dft_aux g (direct_succ_list g x ++ stack) (x :: visited), x_1 = x ∨ (x_1 ∈ reachable_from_list g (direct_succ_list g x) ∨ x_1 ∈ reachable_from_list g stack) ∨ x_1 ∈ visited ⊢ ((x ∈ reachable_from_list g [x] ∨ x ∈ reachable_from_list g stack) ∨ x ∈ visited) ∧ ∀ (a : Node), (a ∈ reachable_from_list g (direct_succ_list g x) ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited → (a ∈ reachable_from_list g [x] ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	simp only [list_direct_succ_set] at h1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : {y \| ∃ x ∈ xs, ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	simp at h1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : {y \| ∃ x ∈ xs, ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	simp only [Set.subset_def]	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	simp	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ ↑xs.toFinset	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	intro a a1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ xs	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a : α a1 : a ∈ reachable_from_list g xs ⊢ a ∈ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	induction a1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a : α a1 : a ∈ reachable_from_list g xs ⊢ a ∈ xs	case base α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α a✝ : x✝ ∈ xs ⊢ x✝ ∈ xs case step α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α e✝ : α × List α a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g xs e✝.1 a_ih✝ : e✝.1 ∈ xs ⊢ x✝ ∈ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	case _ _ ih => exact ih	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α ih : x✝ ∈ xs ⊢ x✝ ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	case _ x e ih_1 ih_2 _ ih_4 => exact h1 x e.fst ih_4 e.snd ih_2 ih_1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x : α e : α × List α ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ xs ⊢ x ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	exact ih	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α ih : x✝ ∈ xs ⊢ x✝ ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_5	[812, 1]	[830, 44]	exact h1 x e.fst ih_4 e.snd ih_2 ih_1	α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x : α e : α × List α ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ xs ⊢ x ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_reachable_from_list	[833, 1]	[843, 32]	apply Set.eq_of_subset_of_subset	α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset = reachable_from_list g xs	case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_reachable_from_list	[833, 1]	[843, 32]	apply base_of_reachable_from_list_is_subset_of_reachable_from_list	case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_reachable_from_list	[833, 1]	[843, 32]	exact extracted_5 g xs h1	case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	have s1 : (dft_aux g stack []).toFinset.toSet = reachable_from_list g (dft_aux g stack []) := by apply list_direct_succ_set_closed_reachable_from_list g (dft_aux g stack []) exact list_direct_succ_set_closed_dft_aux g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	simp only [s1]	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ reachable_from_list g (dft_aux g stack [])
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	apply reachable_from_list_mono g stack (dft_aux g stack [])	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ reachable_from_list g (dft_aux g stack [])	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ stack ⊆ dft_aux g stack []
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	exact stack_subset_dft_aux g stack []	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ stack ⊆ dft_aux g stack []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	apply list_direct_succ_set_closed_reachable_from_list g (dft_aux g stack [])	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack [])	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g (dft_aux g stack []) ⊆ ↑(dft_aux g stack []).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_closed_dft_aux	[846, 1]	[860, 42]	exact list_direct_succ_set_closed_dft_aux g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g (dft_aux g stack []) ⊆ ↑(dft_aux g stack []).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_eq_reachable_from_list	[866, 1]	[878, 43]	have s1 : (dft_aux g stack []).toFinset.toSet ⊆ reachable_from_list g stack ∪ [].toFinset.toSet := dft_aux_is_subset_of_reachable_from_list_and_visited g stack []	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ∪ ↑[].toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_eq_reachable_from_list	[866, 1]	[878, 43]	simp only [List.toFinset_nil, Finset.coe_empty, Set.union_empty] at s1	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ∪ ↑[].toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_eq_reachable_from_list	[866, 1]	[878, 43]	have s2 : reachable_from_list g stack ⊆ (dft_aux g stack []).toFinset.toSet := reachable_from_list_closed_dft_aux g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack s2 : reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_aux_eq_reachable_from_list	[866, 1]	[878, 43]	exact Set.eq_of_subset_of_subset s1 s2	Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack s2 : reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	simp at this	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : (fun x => s' ∈ x) ↑(dft_aux g S []).toFinset = (fun x => s' ∈ x) (reachable_from_list g S) ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	simp [dft, this]	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	clear this	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	constructor	α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S → ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s' case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ (∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s') → s' ∈ reachable_from_list g S
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	intro h	case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S → ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α h : s' ∈ reachable_from_list g S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	induction h with \| base _ h => exact ⟨_, h, .refl⟩ \| step _ _ h1 h2 _ ih => have ⟨_, h3, h4⟩ := ih exact ⟨_, h3, .tail h4 ⟨_, h1, h2⟩⟩	case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α h : s' ∈ reachable_from_list g S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	exact ⟨_, h, .refl⟩	case mp.base α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α h : x✝ ∈ S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	have ⟨_, h3, h4⟩ := ih	case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝	case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 w✝ : α h3 : w✝ ∈ S h4 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) w✝ e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	exact ⟨_, h3, .tail h4 ⟨_, h1, h2⟩⟩	case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 w✝ : α h3 : w✝ ∈ S h4 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) w✝ e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	dft_iff	[884, 1]	[900, 50]	intro ⟨s, h1, h2⟩	case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ (∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s') → s' ∈ reachable_from_list g S	case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S h2 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s' ⊢ s' ∈ reachable_from_list g S

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

apply reachable_from_list_mono g xs (xs ++ ys)

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g (xs ++ ys)

case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ xs ⊆ xs ++ ys case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

simp

case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ xs ⊆ xs ++ ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

exact left

case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node left : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

apply reachable_from_list_mono g ys (xs ++ ys)

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)

case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ ys ⊆ xs ++ ys case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

simp

case h1 Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ ys ⊆ xs ++ ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

exact right

case a Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node right : x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append

[695, 1]

[704, 48]

apply Set.eq_of_subset_of_subset

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) = reachable_from_list g xs ∪ reachable_from_list g ys

case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) ⊆ reachable_from_list g xs ∪ reachable_from_list g ys case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g xs ∪ reachable_from_list g ys ⊆ reachable_from_list g (xs ++ ys)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append

[695, 1]

[704, 48]

apply reachable_from_list_of_append_left

case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g (xs ++ ys) ⊆ reachable_from_list g xs ∪ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append

[695, 1]

[704, 48]

apply reachable_from_list_of_append_right

case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node ⊢ reachable_from_list g xs ∪ reachable_from_list g ys ⊆ reachable_from_list g (xs ++ ys)

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_cons

[707, 1]

[716, 9]

simp only [← reachable_from_list_of_append]

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g [x] ∪ reachable_from_list g ys

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g ([x] ++ ys)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_cons

[707, 1]

[716, 9]

simp

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ys : List Node ⊢ reachable_from_list g (x :: ys) = reachable_from_list g ([x] ++ ys)

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

simp only [Set.subset_def]

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x]

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

intro a a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x]

Node : Type inst✝ : DecidableEq Node g : Graph Node x a : Node a1 : a ∈ reachable_from_list g (direct_succ_list g x) ⊢ a ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

induction a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x a : Node a1 : a ∈ reachable_from_list g (direct_succ_list g x) ⊢ a ∈ reachable_from_list g [x]

case base Node : Type inst✝ : DecidableEq Node g : Graph Node x a x✝ : Node a✝ : x✝ ∈ direct_succ_list g x ⊢ x✝ ∈ reachable_from_list g [x] case step Node : Type inst✝ : DecidableEq Node g : Graph Node x a x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g (direct_succ_list g x) e✝.1 a_ih✝ : e✝.1 ∈ reachable_from_list g [x] ⊢ x✝ ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

case _ x y ih => obtain s1 := mem_direct_succ_list_iff g x y simp only [s1] at ih apply Exists.elim ih intro ys a2 cases a2 case _ a2_left a2_right => apply reachable_from_list.step y (x, ys) a2_right a2_left apply reachable_from_list.base x simp

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x ⊢ y ∈ reachable_from_list g [x]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

case _ x e ih_1 ih_2 _ ih_4 => exact reachable_from_list.step x e ih_1 ih_2 ih_4

Node : Type inst✝ : DecidableEq Node g : Graph Node x✝ a x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g (direct_succ_list g x✝) e.1 ih_4 : e.1 ∈ reachable_from_list g [x✝] ⊢ x ∈ reachable_from_list g [x✝]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

obtain s1 := mem_direct_succ_list_iff g x y

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x ⊢ y ∈ reachable_from_list g [x]

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

simp only [s1] at ih

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node ih : y ∈ direct_succ_list g x s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

apply Exists.elim ih

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ ∀ (a : List Node), y ∈ a ∧ (x, a) ∈ g → y ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

intro ys a2

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ⊢ ∀ (a : List Node), y ∈ a ∧ (x, a) ∈ g → y ∈ reachable_from_list g [x]

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2 : y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

cases a2

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2 : y ∈ ys ∧ (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

case intro Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node left✝ : y ∈ ys right✝ : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

case _ a2_left a2_right => apply reachable_from_list.step y (x, ys) a2_right a2_left apply reachable_from_list.base x simp

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

apply reachable_from_list.step y (x, ys) a2_right a2_left

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ y ∈ reachable_from_list g [x]

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ reachable_from_list g [x] (x, ys).1

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

apply reachable_from_list.base x

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ reachable_from_list g [x] (x, ys).1

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ x ∈ [x]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

simp

Node : Type x✝ : DecidableEq Node g : Graph Node x a y : Node s1 : y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ih : ∃ ys, y ∈ ys ∧ (x, ys) ∈ g ys : List Node a2_left : y ∈ ys a2_right : (x, ys) ∈ g ⊢ x ∈ [x]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list

[719, 1]

[740, 56]

exact reachable_from_list.step x e ih_1 ih_2 ih_4

Node : Type inst✝ : DecidableEq Node g : Graph Node x✝ a x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g (direct_succ_list g x✝) e.1 ih_4 : e.1 ∈ reachable_from_list g [x✝] ⊢ x ∈ reachable_from_list g [x✝]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp only [Set.subset_def]

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ reachable_from_list g (list_direct_succ_list g xs) ⊆ reachable_from_list g xs

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ reachable_from_list g (list_direct_succ_list g xs), x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

intro x a1

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ reachable_from_list g (list_direct_succ_list g xs), x ∈ reachable_from_list g xs

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g xs) ⊢ x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

induction xs

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g xs) ⊢ x ∈ reachable_from_list g xs

case nil Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g [] case cons Node : Type inst✝ : DecidableEq Node g : Graph Node x head✝ : Node tail✝ : List Node tail_ih✝ : x ∈ reachable_from_list g (list_direct_succ_list g tail✝) → x ∈ reachable_from_list g tail✝ a1 : x ∈ reachable_from_list g (list_direct_succ_list g (head✝ :: tail✝)) ⊢ x ∈ reachable_from_list g (head✝ :: tail✝)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

case nil => simp only [list_direct_succ_list] at a1 simp at a1 exact a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g []

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

case cons hd tl ih => simp only [list_direct_succ_list_cons] at a1 simp only [reachable_from_list_of_cons g hd tl] simp simp only [reachable_from_list_of_append] at a1 simp at a1 cases a1 case _ left => left obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd exact Set.mem_of_subset_of_mem s1 left case _ right => right exact ih right

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (list_direct_succ_list g (hd :: tl)) ⊢ x ∈ reachable_from_list g (hd :: tl)

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp only [list_direct_succ_list] at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (list_direct_succ_list g []) ⊢ x ∈ reachable_from_list g []

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (List.map (fun x => direct_succ_list g x) []).join ⊢ x ∈ reachable_from_list g []

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g (List.map (fun x => direct_succ_list g x) []).join ⊢ x ∈ reachable_from_list g []

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g [] ⊢ x ∈ reachable_from_list g []

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

exact a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x : Node a1 : x ∈ reachable_from_list g [] ⊢ x ∈ reachable_from_list g []

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp only [list_direct_succ_list_cons] at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (list_direct_succ_list g (hd :: tl)) ⊢ x ∈ reachable_from_list g (hd :: tl)

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g (hd :: tl)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp only [reachable_from_list_of_cons g hd tl]

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g (hd :: tl)

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∪ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∪ reachable_from_list g tl

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp only [reachable_from_list_of_append] at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd ++ list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∪ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

simp at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∪ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∨ x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

cases a1

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl a1 : x ∈ reachable_from_list g (direct_succ_list g hd) ∨ x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

case inl Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl h✝ : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl case inr Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl h✝ : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

case _ left => left obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd exact Set.mem_of_subset_of_mem s1 left

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

case _ right => right exact ih right

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

left

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

obtain s1 := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g hd

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) ⊢ x ∈ reachable_from_list g [hd]

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) s1 : reachable_from_list g (direct_succ_list g hd) ⊆ reachable_from_list g [hd] ⊢ x ∈ reachable_from_list g [hd]

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

exact Set.mem_of_subset_of_mem s1 left

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl left : x ∈ reachable_from_list g (direct_succ_list g hd) s1 : reachable_from_list g (direct_succ_list g hd) ⊆ reachable_from_list g [hd] ⊢ x ∈ reachable_from_list g [hd]

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

right

Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g [hd] ∨ x ∈ reachable_from_list g tl

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g tl

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_list_direct_succ_list_is_subset_of_reachable_from_list

[743, 1]

[771, 23]

exact ih right

case h Node : Type inst✝ : DecidableEq Node g : Graph Node x hd : Node tl : List Node ih : x ∈ reachable_from_list g (list_direct_succ_list g tl) → x ∈ reachable_from_list g tl right : x ∈ reachable_from_list g (list_direct_succ_list g tl) ⊢ x ∈ reachable_from_list g tl

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

induction stack, visited using dft_aux.induct g

Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset

case case1 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node ⊢ ↑(dft_aux g [] visited✝).toFinset ⊆ reachable_from_list g [] ∪ ↑visited✝.toFinset case case2 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∈ visited✝ ih1✝ : ↑(dft_aux g xs✝ visited✝).toFinset ⊆ reachable_from_list g xs✝ ∪ ↑visited✝.toFinset ⊢ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset ⊆ reachable_from_list g (x✝ :: xs✝) ∪ ↑visited✝.toFinset case case3 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∉ visited✝ ih1✝ : ↑(dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)).toFinset ⊆ reachable_from_list g (direct_succ_list g x✝ ++ xs✝) ∪ ↑(x✝ :: visited✝).toFinset ⊢ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset ⊆ reachable_from_list g (x✝ :: xs✝) ∪ ↑visited✝.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

case _ visited => simp only [dft_aux] simp

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑(dft_aux g [] visited).toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

case _ visited x stack c1 ih => simp only [reachable_from_list_of_cons g x stack] simp only [dft_aux] simp only [if_pos c1] simp only [Set.union_assoc ] exact Set.subset_union_of_subset_right ih (reachable_from_list g [x])

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑(dft_aux g [] visited).toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑visited.toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ ↑visited.toFinset ⊆ reachable_from_list g [] ∪ ↑visited.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [reachable_from_list_of_cons g x stack]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [if_pos c1]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [Set.union_assoc ]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ (reachable_from_list g stack ∪ ↑visited.toFinset)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

exact Set.subset_union_of_subset_right ih (reachable_from_list g [x])

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g stack ∪ ↑visited.toFinset ⊢ ↑(dft_aux g stack visited).toFinset ⊆ reachable_from_list g [x] ∪ (reachable_from_list g stack ∪ ↑visited.toFinset)

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (x :: stack) visited).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [if_neg c1]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [reachable_from_list_of_append g (direct_succ_list g x) stack] at ih

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x ++ stack) ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [reachable_from_list_of_cons g x stack]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (x :: stack) ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

trans (reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ (x :: visited).toFinset.toSet)

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

exact ih

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

have s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] := reachable_from_list_direct_succ_list_is_subset_of_reachable_from_list g x

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp only [Set.subset_def] at *

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset ⊆ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : reachable_from_list g (direct_succ_list g x) ⊆ reachable_from_list g [x] ⊢ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset ⊆ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ∀ x_1 ∈ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset, x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset, x_1 ∈ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

simp at *

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : ∀ x_1 ∈ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset, x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ⊢ ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x) ∪ reachable_from_list g stack ∪ ↑(x :: visited).toFinset, x_1 ∈ reachable_from_list g [x] ∪ reachable_from_list g stack ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ih : ∀ x_1 ∈ dft_aux g (direct_succ_list g x ++ stack) (x :: visited), x_1 = x ∨ (x_1 ∈ reachable_from_list g (direct_succ_list g x) ∨ x_1 ∈ reachable_from_list g stack) ∨ x_1 ∈ visited ⊢ ((x ∈ reachable_from_list g [x] ∨ x ∈ reachable_from_list g stack) ∨ x ∈ visited) ∧ ∀ (a : Node), (a ∈ reachable_from_list g (direct_succ_list g x) ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited → (a ∈ reachable_from_list g [x] ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_is_subset_of_reachable_from_list_and_visited

[774, 1]

[809, 14]

tauto

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited s1 : ∀ x_1 ∈ reachable_from_list g (direct_succ_list g x), x_1 ∈ reachable_from_list g [x] ih : ∀ x_1 ∈ dft_aux g (direct_succ_list g x ++ stack) (x :: visited), x_1 = x ∨ (x_1 ∈ reachable_from_list g (direct_succ_list g x) ∨ x_1 ∈ reachable_from_list g stack) ∨ x_1 ∈ visited ⊢ ((x ∈ reachable_from_list g [x] ∨ x ∈ reachable_from_list g stack) ∨ x ∈ visited) ∧ ∀ (a : Node), (a ∈ reachable_from_list g (direct_succ_list g x) ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited → (a ∈ reachable_from_list g [x] ∨ a ∈ reachable_from_list g stack) ∨ a ∈ visited

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

simp only [list_direct_succ_set] at h1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : {y | ∃ x ∈ xs, ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

simp at h1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : {y | ∃ x ∈ xs, ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

simp only [Set.subset_def]

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

simp

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ ↑xs.toFinset

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

intro a a1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ xs

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a : α a1 : a ∈ reachable_from_list g xs ⊢ a ∈ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

induction a1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a : α a1 : a ∈ reachable_from_list g xs ⊢ a ∈ xs

case base α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α a✝ : x✝ ∈ xs ⊢ x✝ ∈ xs case step α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α e✝ : α × List α a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g xs e✝.1 a_ih✝ : e✝.1 ∈ xs ⊢ x✝ ∈ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

case _ _ ih => exact ih

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α ih : x✝ ∈ xs ⊢ x✝ ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

case _ x e ih_1 ih_2 _ ih_4 => exact h1 x e.fst ih_4 e.snd ih_2 ih_1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x : α e : α × List α ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ xs ⊢ x ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

exact ih

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x✝ : α ih : x✝ ∈ xs ⊢ x✝ ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_5

[812, 1]

[830, 44]

exact h1 x e.fst ih_4 e.snd ih_2 ih_1

α : Type inst : DecidableEq α g : Graph α xs : List α h1 : ∀ (a x : α), x ∈ xs → ∀ (x_1 : List α), a ∈ x_1 → (x, x_1) ∈ g → a ∈ xs a x : α e : α × List α ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ xs ⊢ x ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_reachable_from_list

[833, 1]

[843, 32]

apply Set.eq_of_subset_of_subset

α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset = reachable_from_list g xs

case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_reachable_from_list

[833, 1]

[843, 32]

apply base_of_reachable_from_list_is_subset_of_reachable_from_list

case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_reachable_from_list

[833, 1]

[843, 32]

exact extracted_5 g xs h1

case a α : Type inst✝ : DecidableEq α g : Graph α xs : List α h1 : list_direct_succ_set g xs ⊆ ↑xs.toFinset ⊢ reachable_from_list g xs ⊆ ↑xs.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

have s1 : (dft_aux g stack []).toFinset.toSet = reachable_from_list g (dft_aux g stack []) := by apply list_direct_succ_set_closed_reachable_from_list g (dft_aux g stack []) exact list_direct_succ_set_closed_dft_aux g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

simp only [s1]

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ reachable_from_list g (dft_aux g stack [])

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

apply reachable_from_list_mono g stack (dft_aux g stack [])

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ reachable_from_list g stack ⊆ reachable_from_list g (dft_aux g stack [])

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ stack ⊆ dft_aux g stack []

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

exact stack_subset_dft_aux g stack []

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack []) ⊢ stack ⊆ dft_aux g stack []

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

apply list_direct_succ_set_closed_reachable_from_list g (dft_aux g stack [])

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g (dft_aux g stack [])

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g (dft_aux g stack []) ⊆ ↑(dft_aux g stack []).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_closed_dft_aux

[846, 1]

[860, 42]

exact list_direct_succ_set_closed_dft_aux g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g (dft_aux g stack []) ⊆ ↑(dft_aux g stack []).toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_eq_reachable_from_list

[866, 1]

[878, 43]

have s1 : (dft_aux g stack []).toFinset.toSet ⊆ reachable_from_list g stack ∪ [].toFinset.toSet := dft_aux_is_subset_of_reachable_from_list_and_visited g stack []

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ∪ ↑[].toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_eq_reachable_from_list

[866, 1]

[878, 43]

simp only [List.toFinset_nil, Finset.coe_empty, Set.union_empty] at s1

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ∪ ↑[].toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_eq_reachable_from_list

[866, 1]

[878, 43]

have s2 : reachable_from_list g stack ⊆ (dft_aux g stack []).toFinset.toSet := reachable_from_list_closed_dft_aux g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack s2 : reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_aux_eq_reachable_from_list

[866, 1]

[878, 43]

exact Set.eq_of_subset_of_subset s1 s2

Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node s1 : ↑(dft_aux g stack []).toFinset ⊆ reachable_from_list g stack s2 : reachable_from_list g stack ⊆ ↑(dft_aux g stack []).toFinset ⊢ ↑(dft_aux g stack []).toFinset = reachable_from_list g stack

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

simp at this

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : (fun x => s' ∈ x) ↑(dft_aux g S []).toFinset = (fun x => s' ∈ x) (reachable_from_list g S) ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

simp [dft, this]

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ dft g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

clear this

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α this : s' ∈ dft_aux g S [] ↔ s' ∈ reachable_from_list g S ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

constructor

α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S ↔ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S → ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s' case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ (∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s') → s' ∈ reachable_from_list g S

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

intro h

case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ s' ∈ reachable_from_list g S → ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α h : s' ∈ reachable_from_list g S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

induction h with | base _ h => exact ⟨_, h, .refl⟩ | step _ _ h1 h2 _ ih => have ⟨_, h3, h4⟩ := ih exact ⟨_, h3, .tail h4 ⟨_, h1, h2⟩⟩

case mp α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α h : s' ∈ reachable_from_list g S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

exact ⟨_, h, .refl⟩

case mp.base α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α h : x✝ ∈ S ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

have ⟨_, h3, h4⟩ := ih

case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝

case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 w✝ : α h3 : w✝ ∈ S h4 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) w✝ e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

exact ⟨_, h3, .tail h4 ⟨_, h1, h2⟩⟩

case mp.step α : Type inst✝ : DecidableEq α g : Graph α S : List α s' x✝ : α e✝ : α × List α h1 : e✝ ∈ g h2 : x✝ ∈ e✝.2 a✝ : reachable_from_list g S e✝.1 ih : ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s e✝.1 w✝ : α h3 : w✝ ∈ S h4 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) w✝ e✝.1 ⊢ ∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s x✝

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

dft_iff

[884, 1]

[900, 50]

intro ⟨s, h1, h2⟩

case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' : α ⊢ (∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s') → s' ∈ reachable_from_list g S

case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S h2 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s' ⊢ s' ∈ reachable_from_list g S