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
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