theory Prefix_subtract
  imports Main List_Prefix
begin

section {* A small theory of prefix subtraction *}

text {*
  The notion of @{text "prefix_subtract"} is need to make proofs more readable.
  *}

fun prefix_subtract :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
where
  "prefix_subtract []     xs     = []" 
| "prefix_subtract (x#xs) []     = x#xs" 
| "prefix_subtract (x#xs) (y#ys) = (if x = y then prefix_subtract xs ys else (x#xs))"

lemma [simp]: "(x @ y) - x = y"
apply (induct x)
by (case_tac y, simp+)

lemma [simp]: "x - x = []"
by (induct x, auto)

lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
by (induct x, auto)

lemma [simp]: "x - [] = x"
by (induct x, auto)

lemma [simp]: "(x - y = []) \<Longrightarrow> (x \<le> y)"
proof-   
  have "\<exists>xa. x = xa @ (x - y) \<and> xa \<le> y"
    apply (rule prefix_subtract.induct[of _ x y], simp+)
    by (clarsimp, rule_tac x = "y # xa" in exI, simp+)
  thus "(x - y = []) \<Longrightarrow> (x \<le> y)" by simp
qed

lemma diff_prefix:
  "\<lbrakk>c \<le> a - b; b \<le> a\<rbrakk> \<Longrightarrow> b @ c \<le> a"
by (auto elim:prefixE)

lemma diff_diff_appd: 
  "\<lbrakk>c < a - b; b < a\<rbrakk> \<Longrightarrow> (a - b) - c = a - (b @ c)"
apply (clarsimp simp:strict_prefix_def)
by (drule diff_prefix, auto elim:prefixE)

lemma app_eq_cases[rule_format]:
  "\<forall> x . x @ y = m @ n \<longrightarrow> (x \<le> m \<or> m \<le> x)"
apply (induct y, simp)
apply (clarify, drule_tac x = "x @ [a]" in spec)
by (clarsimp, auto simp:prefix_def)

lemma app_eq_dest:
  "x @ y = m @ n \<Longrightarrow> 
               (x \<le> m \<and> (m - x) @ n = y) \<or> (m \<le> x \<and> (x - m) @ y = n)"
by (frule_tac app_eq_cases, auto elim:prefixE)

end