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theory Defs
imports Main
begin
type_synonym intervals = "(nat*nat) list"
fun inv' :: "nat \<Rightarrow> intervals \<Rightarrow> bool" where
"inv' x [] = True"
| "inv' x ((l,r)#ins) = (x\<le>l \<and> l\<le>r \<and> inv' (Suc (Suc r)) ins)"
definition inv where "inv \<equiv> inv' 0"
fun set_of :: "intervals \<Rightarrow> nat set" where
"set_of [] = {}"
| "set_of ((l,r)#ins) = {l..r} \<union> set_of ins"
fun merge_fst2 where
"merge_fst2 [] = []"
| "merge_fst2 [i] = [i]"
| "merge_fst2 ((l1,r1)#(l2,r2)#ins) = (if Suc r1 = l2 then (l1,r2)#ins else ((l1,r1)#(l2,r2)#ins))"
lemma [simp]: "\<lbrakk>a\<le>b; inv' (Suc b) is\<rbrakk> \<Longrightarrow> set_of (merge_fst2 ((a,b)#is)) = {a..b} \<union> set_of is"
apply (cases "is" rule: merge_fst2.cases)
apply (auto split: if_splits)
done
consts del :: "nat \<Rightarrow> intervals \<Rightarrow> intervals"
consts addi :: "nat \<Rightarrow> nat \<Rightarrow> intervals \<Rightarrow> intervals"
end
theory Submission
imports Defs
begin
fun del :: "nat \<Rightarrow> intervals \<Rightarrow> intervals" where
"del _ _ = []"
lemma del_correct_1: "inv is \<Longrightarrow> inv (del x is)"
sorry
lemma del_correct_2: "inv is \<Longrightarrow> set_of (del x is) = (set_of is) - {x}"
sorry
fun addi :: "nat \<Rightarrow> nat \<Rightarrow> intervals \<Rightarrow> intervals" where
"addi _ _ _ = []"
lemma addi_correct_1: "inv is \<Longrightarrow> i\<le>j \<Longrightarrow> inv (addi i j is)"
sorry
lemma addi_correct_2:
"inv is \<Longrightarrow> i\<le>j \<Longrightarrow> set_of (addi i j is) = {i..j} \<union> (set_of is)"
sorry
end
theory Check
imports Submission
begin
lemma del_correct_1: "inv is \<Longrightarrow> inv (del x is)"
by (rule Submission.del_correct_1)
lemma del_correct_2: "inv is \<Longrightarrow> set_of (del x is) = (set_of is) - {x}"
by (rule Submission.del_correct_2)
lemma addi_correct_1: "inv is \<Longrightarrow> i\<le>j \<Longrightarrow> inv (addi i j is)"
by (rule Submission.addi_correct_1)
lemma addi_correct_2: "inv is \<Longrightarrow> i\<le>j \<Longrightarrow> set_of (addi i j is) = {i..j} \<union> (set_of is)"
by (rule Submission.addi_correct_2)
end