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lib: some sym_heap lemmas regarding heap updates 

Signed-off-by: Michael McInerney <michael.mcinerney@proofcraft.systems>
This commit is contained in:
Michael McInerney 2024-01-10 00:56:15 +10:30 committed by Achim D. Brucker
parent c3d54ae139
commit 57f60084e8
1 changed files with 32 additions and 0 deletions
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32
lib/Heap_List.thy
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@ -172,6 +172,38 @@ lemma sym_heap_symmetric:
lemma sym_heap_None:
"\<lbrakk>sym_heap hp hp'; hp p = None\<rbrakk> \<Longrightarrow> \<forall>p'. hp' p' \<noteq> Some p" unfolding sym_heap_def by force
lemma sym_heap_remove_only:
"\<lbrakk>sym_heap hp hp'; hp' y = Some x\<rbrakk> \<Longrightarrow> sym_heap (hp(x := None)) (hp'(y := None))"
apply (clarsimp simp: sym_heap_def)
apply (metis option.inject)
done
lemma sym_heap_remove_only':
"\<lbrakk>sym_heap hp hp'; hp y = Some x\<rbrakk> \<Longrightarrow> sym_heap (hp(y := None)) (hp'(x := None))"
apply (clarsimp simp: sym_heap_def)
apply (metis option.inject)
done
lemma sym_heap_remove_middle_from_chain:
"\<lbrakk>sym_heap hp hp'; before \<noteq> middle; middle \<noteq> after;
hp before = Some middle; hp middle = Some after\<rbrakk>
\<Longrightarrow> sym_heap (hp(before := Some after, middle := None))
(hp'(after := Some before, middle := None))"
apply (clarsimp simp: sym_heap_def)
apply (metis option.simps(1))
done
lemma sym_heap_connect:
"\<lbrakk>sym_heap hp hp'; hp a = None; hp' b = None \<rbrakk> \<Longrightarrow> sym_heap (hp(a \<mapsto> b)) (hp'(b \<mapsto> a))"
by (force simp: sym_heap_def)
lemma sym_heap_insert_into_middle_of_chain:
"\<lbrakk>sym_heap hp hp'; hp before = Some after; hp middle = None; hp' middle = None\<rbrakk>
\<Longrightarrow> sym_heap (hp(before \<mapsto> middle, middle \<mapsto> after)) (hp'(after \<mapsto> middle, middle \<mapsto> before))"
apply (clarsimp simp: sym_heap_def)
apply (metis option.simps)
done
lemma sym_heap_path_reverse:
"sym_heap hp hp' \<Longrightarrow>
heap_path hp (Some p) (p#ps) (Some p')