full_collection/autth.jsonl

{"id":"def:mygrammar","label":"mygrammar","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define the context-free grammar \\(G_1\\) as \\(S \\to \\varepsilon | a S b\\).","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/mygrammar","latex_link":"sect0002.html#def:mygrammar","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/mygrammar","lean_decl_name":"mygrammar","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"def mygrammar : ( ContextFreeGrammar Alphabet ) where\n  NT := Nonterminals\n  initial := Nonterminals.S\n  rules := [ myfirstrule, mysecondrule ]\n","line":53,"end_line":56,"proof_status":"formalized"}
{"id":"thm:my_second_theorem","label":"my_second_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\varepsilon \\in L(G_1)\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_second_theorem","latex_link":"sect0002.html#thm:my_second_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_second_theorem","lean_decl_name":"my_second_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_second_theorem : List.nil ∈ ( ContextFreeGrammar.language mygrammar ) := by\n  apply (ContextFreeGrammar.mem_language_iff mygrammar []).mpr\n  apply ContextFreeGrammar.Produces.single\n  use myfirstrule\n  constructor\n  · unfold mygrammar\n    simp\n  · refine (ContextFreeRule.rewrites_iff [Symbol.nonterminal mygrammar.initial]\n      (List.map Symbol.terminal [])).mpr\n      ?h.right.a\n    use [], []\n    simp\n    unfold mygrammar myfirstrule\n    simp\n","line":66,"end_line":79,"proof_status":"formalized"}
{"id":"thm:my_first_theorem","label":"my_first_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(aab \\in a^* b^*\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_first_theorem","latex_link":"sect0001.html#thm:my_first_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_first_theorem","lean_decl_name":"my_first_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_first_theorem: [a,a,b] ∈ matches' ( comp ( star ( char a ) ) ( star ( char b ) ) ) := by\n  simp\n  refine Language.mem_mul.mpr ?_\n  use [a,a]\n  constructor\n  refine Language.mem_kstar_iff_exists_nonempty.mpr ?h.left.a\n  use [[a],[a]]\n  simp\n  rfl\n  use [b]\n  constructor\n  refine Language.mem_kstar_iff_exists_nonempty.mpr ?_\n  use [[b]]\n  simp only [List.join_cons, List.join_nil, List.singleton_append, List.mem_singleton, ne_eq,\n    forall_eq, List.cons_ne_self, not_false_eq_true, and_true, true_and]\n  rfl\n  simp\n","line":15,"end_line":31,"proof_status":"formalized"}
{"id":"thm:my_third_theorem","label":"my_third_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(ab \\in L(G_1)\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_third_theorem","latex_link":"sect0002.html#thm:my_third_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_third_theorem","lean_decl_name":"my_third_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_third_theorem : [a,b] ∈ ( ContextFreeGrammar.language mygrammar ) := by\n  apply (ContextFreeGrammar.mem_language_iff mygrammar [a, b]).mpr\n  have fst : mygrammar.Derives [Symbol.nonterminal mygrammar.initial]\n    [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.nonterminal  Nonterminals.S ),\n    ( Symbol.terminal Alphabet.b ) ] := by\n    apply ContextFreeGrammar.Produces.single\n    use mysecondrule\n    constructor\n    · unfold mygrammar\n      simp\n    · apply (ContextFreeRule.rewrites_iff [Symbol.nonterminal mygrammar.initial]\n        [Symbol.terminal a, Symbol.nonterminal Nonterminals.S, Symbol.terminal b]).mpr\n      use [], []\n      simp\n      unfold mygrammar mysecondrule\n      simp\n  have snd : mygrammar.Derives [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.nonterminal  Nonterminals.S ),\n    ( Symbol.terminal Alphabet.b ) ] [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.terminal Alphabet.b ) ] := by\n    apply ContextFreeGrammar.Produces.single\n    use myfirstrule\n    constructor\n    · unfold mygrammar\n      simp\n    · apply (ContextFreeRule.rewrites_iff [Symbol.terminal a, Symbol.nonterminal Nonterminals.S, Symbol.terminal b]\n      [Symbol.terminal a, Symbol.terminal b]).mpr\n      use [Symbol.terminal a], [Symbol.terminal b]\n      unfold myfirstrule\n      simp\n  exact ContextFreeGrammar.Derives.trans fst snd\n","line":82,"end_line":113,"proof_status":"formalized"}