Btukfyl

I would like to prove the following lemma in Isabelle:

lemma "T (Open # xs) ⟹ ¬ S (Open # xs) ⟹ count xs Close ≤ count xs Open"

Please find the definitions below:

datatype paren = Open | Close



inductive S where

  S_empty: "S " |

  S_append: "S xs ⟹ S ys ⟹ S (xs @ ys)" |

  S_paren: "S xs ⟹ S (Open # xs @ [Close])"



inductive T where

  T_S: "T " |

  T_append: "T xs ⟹ T ys ⟹ T (xs @ ys)" |

  T_paren: "T xs ⟹ T (Open # xs @ [Close])" |

  T_left: "T xs ⟹ T (Open # xs)"

The lemma states that an unbalanced parentheses structure would result in a possibly unbalanced structure when removing an Open bracket.

I've been trying the techniques that are described in the book "A proof-assistant for Higher-order logic", but so far none of them work. In particular, I tried to use rule inversion and rule induction, sledgehammer and other techniques.

One of the problems is that I haven't yet learned about Isar proofs, which thus complicates the proof. I would prefer if you can orient me with plain apply commands.

Please find a proof below. It is not unlikely that it can be improved: I tried to follow the simplest route towards the proof and relied on sledgehammer to fill in the details.

theory so_raoidii

imports  Complex_Main



begin



declare [[show_types]]

declare [[show_sorts]]



datatype paren = Open | Close



inductive S where

  S_empty: "S " |

  S_append: "S xs ⟹ S ys ⟹ S (xs @ ys)" |

  S_paren: "S xs ⟹ S (Open # xs @ [Close])"



inductive T where

  T_S: "T " |

  T_append: "T xs ⟹ T ys ⟹ T (xs @ ys)" |

  T_paren: "T xs ⟹ T (Open # xs @ [Close])" |

  T_left: "T xs ⟹ T (Open # xs)"



lemma count_list_lem: 

  "count_list xsa a = n ⟹ 

  count_list ysa a = m ⟹ 

  count_list (xsa @ ysa) a = n + m"

  apply(induction xsa arbitrary: ysa n m)

  apply auto

  done



lemma T_to_count: "T xs ⟹ count_list xs Close ≤ count_list xs Open"

  apply(induction rule: T.induct)

  by (simp add: count_list_lem)+



lemma T_to_S_count: "T xs ⟹ count_list xs Close = count_list xs Open ⟹ S xs"

  apply(induction rule: T.induct)

  apply(auto)

  apply(simp add: S_empty)

  apply(metis S_append T_to_count add.commute add_le_cancel_right count_list_lem 

        dual_order.antisym)

  apply(simp add: count_list_lem S_paren)

  using T_to_count by fastforce



lemma "T (Open # xs) ⟹ 

      ¬ S (Open # xs) ⟹ 

      count_list xs Close ≤ count_list xs Open"

  apply(cases "T xs")

  apply(simp add: T_to_count)

  using T_to_S_count T_to_count by fastforce



end