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