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	visited_subset_dft_aux	[388, 1]	[409, 17]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ dft_aux g (x :: stack) visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	visited_subset_dft_aux	[388, 1]	[409, 17]	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 : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	visited_subset_dft_aux	[388, 1]	[409, 17]	trans (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ x :: visited Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	visited_subset_dft_aux	[388, 1]	[409, 17]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ x :: visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	visited_subset_dft_aux	[388, 1]	[409, 17]	exact ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_1	[412, 1]	[424, 15]	simp	α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x :: zs ⊆ xs	α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs ∧ zs ⊆ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_1	[412, 1]	[424, 15]	constructor	α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs ∧ zs ⊆ xs	case left α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs case right α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ zs ⊆ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_1	[412, 1]	[424, 15]	apply Set.mem_of_subset_of_mem h3 h2	case left α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_1	[412, 1]	[424, 15]	exact h1	case right α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ zs ⊆ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_2	[427, 1]	[436, 10]	simp at *	α : Type ws xs ys zs : List α x : α h1 : ws ++ ys ⊆ zs h2 : x :: xs ⊆ zs ⊢ x :: ys ⊆ zs	α : Type ws xs ys zs : List α x : α h1 : ws ⊆ zs ∧ ys ⊆ zs h2 : x ∈ zs ∧ xs ⊆ zs ⊢ x ∈ zs ∧ ys ⊆ zs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_2	[427, 1]	[436, 10]	tauto	α : Type ws xs ys zs : List α x : α h1 : ws ⊆ zs ∧ ys ⊆ zs h2 : x ∈ zs ∧ xs ⊆ zs ⊢ x ∈ zs ∧ ys ⊆ zs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	induction stack, visited using dft_aux.induct g	Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node ⊢ stack ⊆ dft_aux g stack visited	case case1 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node ⊢ [] ⊆ dft_aux g [] visited✝ case case2 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∈ visited✝ ih1✝ : xs✝ ⊆ dft_aux g xs✝ visited✝ ⊢ x✝ :: xs✝ ⊆ dft_aux g (x✝ :: xs✝) visited✝ case case3 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node x✝ : Node xs✝ : List Node h✝ : x✝ ∉ visited✝ ih1✝ : direct_succ_list g x✝ ++ xs✝ ⊆ dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝) ⊢ x✝ :: xs✝ ⊆ dft_aux g (x✝ :: xs✝) visited✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	case _ visited => simp only [dft_aux] simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ dft_aux g [] visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_pos c1] apply extracted_1 (dft_aux g stack visited) visited stack x ih c1 exact visited_subset_dft_aux g stack visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g (x :: stack) visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_neg c1] apply extracted_2 (direct_succ_list g x) visited stack (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) x ih apply visited_subset_dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ dft_aux g (x :: stack) visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ dft_aux g [] visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ visited
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g (x :: stack) visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	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 : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g stack visited
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	apply extracted_1 (dft_aux g stack visited) visited stack x ih c1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g stack visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ visited ⊆ dft_aux g stack visited
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	exact visited_subset_dft_aux g stack visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ visited ⊆ dft_aux g stack visited	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ dft_aux g (x :: stack) visited	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	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 : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	apply extracted_2 (direct_succ_list g x) visited stack (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) x ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	stack_subset_dft_aux	[439, 1]	[460, 84]	apply visited_subset_dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	simp only [list_cons_to_set_union]	α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑(x :: ys).toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	apply Set.union_subset	α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	case sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset case tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	apply Set.union_subset	case sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ⊆ ↑ys.toFinset ∪ ↑xs.toFinset case sr.tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	simp	case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ ys ∨ x ∈ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	right	case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ ys ∨ x ∈ xs	case sr.sr.h α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	exact h1	case sr.sr.h α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	exact Set.subset_union_left ys.toFinset.toSet xs.toFinset.toSet	case sr.tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_3	[463, 1]	[478, 71]	exact Set.subset_union_right ys.toFinset.toSet xs.toFinset.toSet	case tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp only [list_cons_to_set_union] at h1	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ ↑(x :: zs).toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp only [Set.subset_def] at h1	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, x_1 ∈ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp at h1	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, x_1 ∈ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp only [list_cons_to_set_union]	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ({x} ∪ ↑ys.toFinset)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp only [list_append_to_set_union]	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ({x} ∪ ↑ys.toFinset)	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp only [Set.subset_def]	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ x_1 ∈ ↑xs.toFinset ∪ S, x_1 ∈ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	simp	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ x_1 ∈ ↑xs.toFinset ∪ S, x_1 ∈ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ (x_1 : Node), x_1 ∈ xs ∨ x_1 ∈ S → x_1 = x ∨ (x_1 ∈ xs ∨ x_1 ∈ zs) ∨ x_1 ∈ ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	intro a a1	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ (x_1 : Node), x_1 ∈ xs ∨ x_1 ∈ S → x_1 = x ∨ (x_1 ∈ xs ∨ x_1 ∈ zs) ∨ x_1 ∈ ys	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys a : Node a1 : a ∈ xs ∨ a ∈ S ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	specialize h1 a	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys a : Node a1 : a ∈ xs ∨ a ∈ S ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x a : Node a1 : a ∈ xs ∨ a ∈ S h1 : a ∈ S → (a = x ∨ a ∈ zs) ∨ a ∈ ys ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	extracted_4	[481, 1]	[500, 10]	tauto	Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x a : Node a1 : a ∈ xs ∨ a ∈ S h1 : a ∈ S → (a = x ∨ a ∈ zs) ∨ a ∈ ys ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	induction stack, visited using dft_aux.induct g	Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset	case case1 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node h1 : list_direct_succ_set g visited✝ ⊆ ↑[].toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited✝) ⊆ ↑(dft_aux 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✝ : list_direct_succ_set g visited✝ ⊆ ↑xs✝.toFinset ∪ ↑visited✝.toFinset → list_direct_succ_set g (dft_aux g xs✝ visited✝) ⊆ ↑(dft_aux g xs✝ visited✝).toFinset h1 : list_direct_succ_set g visited✝ ⊆ ↑(x✝ :: xs✝).toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g (x✝ :: xs✝) visited✝) ⊆ ↑(dft_aux 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✝ : list_direct_succ_set g (x✝ :: visited✝) ⊆ ↑(direct_succ_list g x✝ ++ xs✝).toFinset ∪ ↑(x✝ :: visited✝).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)) ⊆ ↑(dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)).toFinset h1 : list_direct_succ_set g visited✝ ⊆ ↑(x✝ :: xs✝).toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g (x✝ :: xs✝) visited✝) ⊆ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	case _ visited => simp at h1 simp only [dft_aux] simp exact h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑[].toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_neg c1] apply ih simp only [list_direct_succ_set_cons] exact extracted_4 (direct_succ_list g x) visited stack (list_direct_succ_set g visited) x h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux g (x :: stack) visited).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp at h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑[].toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ {a \| a ∈ visited}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	exact h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a \| a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ {a \| a ∈ visited}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux 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 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	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 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	apply ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	trans (x :: stack).toFinset.toSet ∪ visited.toFinset.toSet	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊆ ↑stack.toFinset ∪ ↑visited.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	exact h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	exact extracted_3 visited stack x c1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊆ ↑stack.toFinset ∪ ↑visited.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp only [dft_aux]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux 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 : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	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 : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	apply ih	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	simp only [list_direct_succ_set_cons]	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(direct_succ_list g x).toFinset ∪ list_direct_succ_set g visited ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux_aux	[503, 1]	[531, 99]	exact extracted_4 (direct_succ_list g x) visited stack (list_direct_succ_set g visited) x h1	Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(direct_succ_list g x).toFinset ∪ list_direct_succ_set g visited ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux	[534, 1]	[543, 9]	apply list_direct_succ_set_closed_dft_aux_aux	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	case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g [] ⊆ ↑stack.toFinset ∪ ↑[].toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux	[534, 1]	[543, 9]	simp only [list_direct_succ_set]	case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g [] ⊆ ↑stack.toFinset ∪ ↑[].toFinset	case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ {y \| ∃ x ∈ [], ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑stack.toFinset ∪ ↑[].toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_direct_succ_set_closed_dft_aux	[534, 1]	[543, 9]	simp	case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ {y \| ∃ x ∈ [], ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑stack.toFinset ∪ ↑[].toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	base_of_reachable_from_list_is_subset_of_reachable_from_list	[613, 1]	[623, 40]	simp only [Set.subset_def]	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ ↑xs.toFinset, x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	base_of_reachable_from_list_is_subset_of_reachable_from_list	[613, 1]	[623, 40]	intro x a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ ↑xs.toFinset, x ∈ reachable_from_list g xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ ↑xs.toFinset ⊢ x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	base_of_reachable_from_list_is_subset_of_reachable_from_list	[613, 1]	[623, 40]	simp at a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ ↑xs.toFinset ⊢ x ∈ reachable_from_list g xs	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	base_of_reachable_from_list_is_subset_of_reachable_from_list	[613, 1]	[623, 40]	exact reachable_from_list.base x a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	simp only [Set.subset_def]	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ reachable_from_list g xs ⊆ reachable_from_list g ys	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	intro x a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ reachable_from_list g ys	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x : Node a1 : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	induction a1	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x : Node a1 : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g ys	case base Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x x✝ : Node a✝ : x✝ ∈ xs ⊢ x✝ ∈ reachable_from_list g ys case step Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g xs e✝.1 a_ih✝ : e✝.1 ∈ 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_mono	[626, 1]	[641, 56]	case base a ih => apply reachable_from_list.base apply Set.mem_of_subset_of_mem h1 ih	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	case step 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 xs ys : List Node h1 : xs ⊆ ys x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ 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_mono	[626, 1]	[641, 56]	apply reachable_from_list.base	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ reachable_from_list g ys	case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	apply Set.mem_of_subset_of_mem h1 ih	case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_mono	[626, 1]	[641, 56]	exact reachable_from_list.step x e ih_1 ih_2 ih_4	Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ 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_left	[644, 1]	[671, 59]	simp only [Set.subset_def]	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	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node ⊢ ∀ x ∈ reachable_from_list g (xs ++ ys), x ∈ reachable_from_list g xs ∪ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	intro x a1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node ⊢ ∀ x ∈ reachable_from_list g (xs ++ ys), x ∈ reachable_from_list g xs ∪ reachable_from_list g ys	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	simp	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	induction a1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case base Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x x✝ : Node a✝ : x✝ ∈ xs ++ ys ⊢ x✝ ∈ reachable_from_list g xs ∨ x✝ ∈ reachable_from_list g ys case step Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g (xs ++ ys) e✝.1 a_ih✝ : e✝.1 ∈ reachable_from_list g xs ∨ e✝.1 ∈ reachable_from_list g ys ⊢ x✝ ∈ reachable_from_list g xs ∨ x✝ ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case _ x ih => simp at ih cases ih case inl c1 => left exact reachable_from_list.base x c1 case inr c1 => right exact reachable_from_list.base x c1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ++ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case _ x e ih_1 ih_2 ih_3 ih_4 => cases ih_4 case _ left => left exact reachable_from_list.step x e ih_1 ih_2 left case _ right => right exact reachable_from_list.step x e ih_1 ih_2 right	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 ih_4 : e.1 ∈ reachable_from_list g xs ∨ e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	simp at ih	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ++ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ∨ x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	cases ih	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ∨ x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case inl Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node h✝ : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys case inr Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node h✝ : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case inl c1 => left exact reachable_from_list.base x c1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case inr c1 => right exact reachable_from_list.base x c1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	left	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	exact reachable_from_list.base x c1	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ 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_left	[644, 1]	[671, 59]	right	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	exact reachable_from_list.base x c1	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ 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_left	[644, 1]	[671, 59]	cases ih_4	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 ih_4 : e.1 ∈ reachable_from_list g xs ∨ e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case inl Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 h✝ : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys case inr Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 h✝ : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case _ left => left exact reachable_from_list.step x e ih_1 ih_2 left	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	case _ right => right exact reachable_from_list.step x e ih_1 ih_2 right	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_left	[644, 1]	[671, 59]	left	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ 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_left	[644, 1]	[671, 59]	exact reachable_from_list.step x e ih_1 ih_2 left	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ 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_left	[644, 1]	[671, 59]	right	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ 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_left	[644, 1]	[671, 59]	exact reachable_from_list.step x e ih_1 ih_2 right	case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ 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_right	[674, 1]	[692, 20]	simp only [Set.subset_def]	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)	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node ⊢ ∀ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys, x ∈ reachable_from_list g (xs ++ ys)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	intro x a1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node ⊢ ∀ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys, x ∈ reachable_from_list g (xs ++ ys)	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∪ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	reachable_from_list_of_append_right	[674, 1]	[692, 20]	simp at a1	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∪ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)	Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

visited_subset_dft_aux

[388, 1]

[409, 17]

simp only [dft_aux]

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

visited_subset_dft_aux

[388, 1]

[409, 17]

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 : x :: visited ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ visited ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

visited_subset_dft_aux

[388, 1]

[409, 17]

trans (x :: visited)

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

visited_subset_dft_aux

[388, 1]

[409, 17]

simp

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

visited_subset_dft_aux

[388, 1]

[409, 17]

exact ih

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_1

[412, 1]

[424, 15]

simp

α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x :: zs ⊆ xs

α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs ∧ zs ⊆ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_1

[412, 1]

[424, 15]

constructor

α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs ∧ zs ⊆ xs

case left α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs case right α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ zs ⊆ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_1

[412, 1]

[424, 15]

apply Set.mem_of_subset_of_mem h3 h2

case left α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ x ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_1

[412, 1]

[424, 15]

exact h1

case right α : Type xs ys zs : List α x : α h1 : zs ⊆ xs h2 : x ∈ ys h3 : ys ⊆ xs ⊢ zs ⊆ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_2

[427, 1]

[436, 10]

simp at *

α : Type ws xs ys zs : List α x : α h1 : ws ++ ys ⊆ zs h2 : x :: xs ⊆ zs ⊢ x :: ys ⊆ zs

α : Type ws xs ys zs : List α x : α h1 : ws ⊆ zs ∧ ys ⊆ zs h2 : x ∈ zs ∧ xs ⊆ zs ⊢ x ∈ zs ∧ ys ⊆ zs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_2

[427, 1]

[436, 10]

tauto

α : Type ws xs ys zs : List α x : α h1 : ws ⊆ zs ∧ ys ⊆ zs h2 : x ∈ zs ∧ xs ⊆ zs ⊢ x ∈ zs ∧ ys ⊆ zs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

induction stack, visited using dft_aux.induct g

Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node ⊢ stack ⊆ dft_aux g stack visited

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

case _ visited => simp only [dft_aux] simp

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ dft_aux g [] visited

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_pos c1] apply extracted_1 (dft_aux g stack visited) visited stack x ih c1 exact visited_subset_dft_aux g stack visited

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_neg c1] apply extracted_2 (direct_succ_list g x) visited stack (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) x ih apply visited_subset_dft_aux g (direct_succ_list g x ++ stack) (x :: visited)

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ dft_aux g [] visited

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ visited

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

simp

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node ⊢ [] ⊆ visited

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

simp only [dft_aux]

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

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 : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g stack visited

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

apply extracted_1 (dft_aux g stack visited) visited stack x ih c1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ x :: stack ⊆ dft_aux g stack visited

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ visited ⊆ dft_aux g stack visited

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

exact visited_subset_dft_aux g stack visited

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : stack ⊆ dft_aux g stack visited ⊢ visited ⊆ dft_aux g stack visited

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

simp only [dft_aux]

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

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 : direct_succ_list g x ++ stack ⊆ dft_aux g (direct_succ_list g x ++ stack) (x :: visited) ⊢ x :: stack ⊆ if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

apply extracted_2 (direct_succ_list g x) visited stack (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) x ih

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

stack_subset_dft_aux

[439, 1]

[460, 84]

apply visited_subset_dft_aux g (direct_succ_list g x ++ stack) (x :: visited)

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

simp only [list_cons_to_set_union]

α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑(x :: ys).toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

apply Set.union_subset

α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ∪ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

case sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset case tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

apply Set.union_subset

case sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ∪ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ⊆ ↑ys.toFinset ∪ ↑xs.toFinset case sr.tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

simp

case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ {x} ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ ys ∨ x ∈ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

right

case sr.sr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ ys ∨ x ∈ xs

case sr.sr.h α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

exact h1

case sr.sr.h α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ x ∈ xs

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

exact Set.subset_union_left ys.toFinset.toSet xs.toFinset.toSet

case sr.tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑ys.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_3

[463, 1]

[478, 71]

exact Set.subset_union_right ys.toFinset.toSet xs.toFinset.toSet

case tr α : Type inst : DecidableEq α xs ys : List α x : α h1 : x ∈ xs ⊢ ↑xs.toFinset ⊆ ↑ys.toFinset ∪ ↑xs.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp only [list_cons_to_set_union] at h1

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ ↑(x :: zs).toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp only [Set.subset_def] at h1

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : S ⊆ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, x_1 ∈ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp at h1

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, x_1 ∈ {x} ∪ ↑zs.toFinset ∪ ↑ys.toFinset ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp only [list_cons_to_set_union]

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ↑(x :: ys).toFinset

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ({x} ∪ ↑ys.toFinset)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp only [list_append_to_set_union]

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑(xs ++ zs).toFinset ∪ ({x} ∪ ↑ys.toFinset)

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp only [Set.subset_def]

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ↑xs.toFinset ∪ S ⊆ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ x_1 ∈ ↑xs.toFinset ∪ S, x_1 ∈ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

simp

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ x_1 ∈ ↑xs.toFinset ∪ S, x_1 ∈ ↑xs.toFinset ∪ ↑zs.toFinset ∪ ({x} ∪ ↑ys.toFinset)

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ (x_1 : Node), x_1 ∈ xs ∨ x_1 ∈ S → x_1 = x ∨ (x_1 ∈ xs ∨ x_1 ∈ zs) ∨ x_1 ∈ ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

intro a a1

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys ⊢ ∀ (x_1 : Node), x_1 ∈ xs ∨ x_1 ∈ S → x_1 = x ∨ (x_1 ∈ xs ∨ x_1 ∈ zs) ∨ x_1 ∈ ys

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys a : Node a1 : a ∈ xs ∨ a ∈ S ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

specialize h1 a

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x : Node h1 : ∀ x_1 ∈ S, (x_1 = x ∨ x_1 ∈ zs) ∨ x_1 ∈ ys a : Node a1 : a ∈ xs ∨ a ∈ S ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x a : Node a1 : a ∈ xs ∨ a ∈ S h1 : a ∈ S → (a = x ∨ a ∈ zs) ∨ a ∈ ys ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

extracted_4

[481, 1]

[500, 10]

tauto

Node : Type inst : DecidableEq Node xs ys zs : List Node S : Set Node x a : Node a1 : a ∈ xs ∨ a ∈ S h1 : a ∈ S → (a = x ∨ a ∈ zs) ∨ a ∈ ys ⊢ a = x ∨ (a ∈ xs ∨ a ∈ zs) ∨ a ∈ ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

induction stack, visited using dft_aux.induct g

Node : Type inst✝ : DecidableEq Node g : Graph Node stack visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset

case case1 Node : Type inst✝ : DecidableEq Node g : Graph Node visited✝ : List Node h1 : list_direct_succ_set g visited✝ ⊆ ↑[].toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited✝) ⊆ ↑(dft_aux 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✝ : list_direct_succ_set g visited✝ ⊆ ↑xs✝.toFinset ∪ ↑visited✝.toFinset → list_direct_succ_set g (dft_aux g xs✝ visited✝) ⊆ ↑(dft_aux g xs✝ visited✝).toFinset h1 : list_direct_succ_set g visited✝ ⊆ ↑(x✝ :: xs✝).toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g (x✝ :: xs✝) visited✝) ⊆ ↑(dft_aux 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✝ : list_direct_succ_set g (x✝ :: visited✝) ⊆ ↑(direct_succ_list g x✝ ++ xs✝).toFinset ∪ ↑(x✝ :: visited✝).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)) ⊆ ↑(dft_aux g (direct_succ_list g x✝ ++ xs✝) (x✝ :: visited✝)).toFinset h1 : list_direct_succ_set g visited✝ ⊆ ↑(x✝ :: xs✝).toFinset ∪ ↑visited✝.toFinset ⊢ list_direct_succ_set g (dft_aux g (x✝ :: xs✝) visited✝) ⊆ ↑(dft_aux g (x✝ :: xs✝) visited✝).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

case _ visited => simp at h1 simp only [dft_aux] simp exact h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑[].toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

case _ visited x stack c1 ih => simp only [dft_aux] simp only [if_neg c1] apply ih simp only [list_direct_succ_set_cons] exact extracted_4 (direct_succ_list g x) visited stack (list_direct_succ_set g visited) x h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux g (x :: stack) visited).toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp at h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ ↑[].toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g (dft_aux g [] visited) ⊆ ↑(dft_aux g [] visited).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ {a | a ∈ visited}

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

exact h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node h1 : list_direct_succ_set g visited ⊆ {a | a ∈ visited} ⊢ list_direct_succ_set g visited ⊆ {a | a ∈ visited}

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux 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 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

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 : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

apply ih

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

trans (x :: stack).toFinset.toSet ∪ visited.toFinset.toSet

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊆ ↑stack.toFinset ∪ ↑visited.toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

exact h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

exact extracted_3 visited stack x c1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∈ visited ih : list_direct_succ_set g visited ⊆ ↑stack.toFinset ∪ ↑visited.toFinset → list_direct_succ_set g (dft_aux g stack visited) ⊆ ↑(dft_aux g stack visited).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊆ ↑stack.toFinset ∪ ↑visited.toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp only [dft_aux]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (x :: stack) visited) ⊆ ↑(dft_aux 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 : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

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 : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(if x ∈ visited then dft_aux g stack visited else dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

apply ih

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

simp only [list_direct_succ_set_cons]

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(direct_succ_list g x).toFinset ∪ list_direct_succ_set g visited ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux_aux

[503, 1]

[531, 99]

exact extracted_4 (direct_succ_list g x) visited stack (list_direct_succ_set g visited) x h1

Node : Type inst✝ : DecidableEq Node g : Graph Node visited : List Node x : Node stack : List Node c1 : x ∉ visited ih : list_direct_succ_set g (x :: visited) ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset → list_direct_succ_set g (dft_aux g (direct_succ_list g x ++ stack) (x :: visited)) ⊆ ↑(dft_aux g (direct_succ_list g x ++ stack) (x :: visited)).toFinset h1 : list_direct_succ_set g visited ⊆ ↑(x :: stack).toFinset ∪ ↑visited.toFinset ⊢ ↑(direct_succ_list g x).toFinset ∪ list_direct_succ_set g visited ⊆ ↑(direct_succ_list g x ++ stack).toFinset ∪ ↑(x :: visited).toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux

[534, 1]

[543, 9]

apply list_direct_succ_set_closed_dft_aux_aux

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

case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g [] ⊆ ↑stack.toFinset ∪ ↑[].toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux

[534, 1]

[543, 9]

simp only [list_direct_succ_set]

case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ list_direct_succ_set g [] ⊆ ↑stack.toFinset ∪ ↑[].toFinset

case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ {y | ∃ x ∈ [], ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑stack.toFinset ∪ ↑[].toFinset

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

list_direct_succ_set_closed_dft_aux

[534, 1]

[543, 9]

simp

case h1 Node : Type inst✝ : DecidableEq Node g : Graph Node stack : List Node ⊢ {y | ∃ x ∈ [], ∃ ys, y ∈ ys ∧ (x, ys) ∈ g} ⊆ ↑stack.toFinset ∪ ↑[].toFinset

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

base_of_reachable_from_list_is_subset_of_reachable_from_list

[613, 1]

[623, 40]

simp only [Set.subset_def]

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ↑xs.toFinset ⊆ reachable_from_list g xs

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ ↑xs.toFinset, x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

base_of_reachable_from_list_is_subset_of_reachable_from_list

[613, 1]

[623, 40]

intro x a1

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node ⊢ ∀ x ∈ ↑xs.toFinset, x ∈ reachable_from_list g xs

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ ↑xs.toFinset ⊢ x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

base_of_reachable_from_list_is_subset_of_reachable_from_list

[613, 1]

[623, 40]

simp at a1

Node : Type inst✝ : DecidableEq Node g : Graph Node xs : List Node x : Node a1 : x ∈ ↑xs.toFinset ⊢ x ∈ reachable_from_list g xs

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

base_of_reachable_from_list_is_subset_of_reachable_from_list

[613, 1]

[623, 40]

exact reachable_from_list.base x a1

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

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

simp only [Set.subset_def]

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ reachable_from_list g xs ⊆ reachable_from_list g ys

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

intro x a1

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys ⊢ ∀ x ∈ reachable_from_list g xs, x ∈ reachable_from_list g ys

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x : Node a1 : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

induction a1

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x : Node a1 : x ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g ys

case base Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x x✝ : Node a✝ : x✝ ∈ xs ⊢ x✝ ∈ reachable_from_list g ys case step Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g xs e✝.1 a_ih✝ : e✝.1 ∈ 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_mono

[626, 1]

[641, 56]

case base a ih => apply reachable_from_list.base apply Set.mem_of_subset_of_mem h1 ih

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

case step 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 xs ys : List Node h1 : xs ⊆ ys x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ 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_mono

[626, 1]

[641, 56]

apply reachable_from_list.base

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ reachable_from_list g ys

case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

apply Set.mem_of_subset_of_mem h1 ih

case a Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x a : Node ih : a ∈ xs ⊢ a ∈ ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_mono

[626, 1]

[641, 56]

exact reachable_from_list.step x e ih_1 ih_2 ih_4

Node : Type inst✝ : DecidableEq Node g : Graph Node xs ys : List Node h1 : xs ⊆ ys x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 a✝ : reachable_from_list g xs e.1 ih_4 : e.1 ∈ 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_left

[644, 1]

[671, 59]

simp only [Set.subset_def]

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

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

intro x a1

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

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

simp

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∪ reachable_from_list g ys

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

induction a1

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g (xs ++ ys) ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case base Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x x✝ : Node a✝ : x✝ ∈ xs ++ ys ⊢ x✝ ∈ reachable_from_list g xs ∨ x✝ ∈ reachable_from_list g ys case step Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x x✝ : Node e✝ : Node × List Node a✝² : e✝ ∈ g a✝¹ : x✝ ∈ e✝.2 a✝ : reachable_from_list g (xs ++ ys) e✝.1 a_ih✝ : e✝.1 ∈ reachable_from_list g xs ∨ e✝.1 ∈ reachable_from_list g ys ⊢ x✝ ∈ reachable_from_list g xs ∨ x✝ ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

case _ x ih => simp at ih cases ih case inl c1 => left exact reachable_from_list.base x c1 case inr c1 => right exact reachable_from_list.base x c1

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ++ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

case _ x e ih_1 ih_2 ih_3 ih_4 => cases ih_4 case _ left => left exact reachable_from_list.step x e ih_1 ih_2 left case _ right => right exact reachable_from_list.step x e ih_1 ih_2 right

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 ih_4 : e.1 ∈ reachable_from_list g xs ∨ e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

simp at ih

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ++ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ∨ x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

cases ih

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node ih : x ∈ xs ∨ x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case inl Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node h✝ : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys case inr Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node h✝ : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

case inl c1 => left exact reachable_from_list.base x c1

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

case inr c1 => right exact reachable_from_list.base x c1

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

left

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ xs ⊢ x ∈ reachable_from_list g xs

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

exact reachable_from_list.base x c1

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ 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_left

[644, 1]

[671, 59]

right

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ ys ⊢ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

exact reachable_from_list.base x c1

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node c1 : x ∈ 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_left

[644, 1]

[671, 59]

cases ih_4

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 ih_4 : e.1 ∈ reachable_from_list g xs ∨ e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case inl Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 h✝ : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys case inr Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 h✝ : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

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

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

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

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

no goals

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_left

[644, 1]

[671, 59]

left

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ reachable_from_list g xs ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ 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_left

[644, 1]

[671, 59]

exact reachable_from_list.step x e ih_1 ih_2 left

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 left : e.1 ∈ 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_left

[644, 1]

[671, 59]

right

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ 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_left

[644, 1]

[671, 59]

exact reachable_from_list.step x e ih_1 ih_2 right

case h Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x✝ x : Node e : Node × List Node ih_1 : e ∈ g ih_2 : x ∈ e.2 ih_3 : reachable_from_list g (xs ++ ys) e.1 right : e.1 ∈ 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_right

[674, 1]

[692, 20]

simp only [Set.subset_def]

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)

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

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

intro x a1

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

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∪ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/DFT.lean

reachable_from_list_of_append_right

[674, 1]

[692, 20]

simp at a1

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∪ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)

Node : Type inst : DecidableEq Node g : Graph Node xs ys : List Node x : Node a1 : x ∈ reachable_from_list g xs ∨ x ∈ reachable_from_list g ys ⊢ x ∈ reachable_from_list g (xs ++ ys)