| {"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"} |
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