(* Title: HOL/Lex/Prefix.thy
ID: $Id$
Author: Richard Mayr & Tobias Nipkow
Copyright 1995 TUM
*)
(** <= is a partial order: **)
Goalw [prefix_def] "xs <= (xs::'a list)";
by (Simp_tac 1);
qed "prefix_refl";
AddIffs[prefix_refl];
Goalw [prefix_def] "!!xs::'a list. [| xs <= ys; ys <= zs |] ==> xs <= zs";
by (Clarify_tac 1);
by (Simp_tac 1);
qed "prefix_trans";
Goalw [prefix_def] "!!xs::'a list. [| xs <= ys; ys <= xs |] ==> xs = ys";
by (Clarify_tac 1);
by (Asm_full_simp_tac 1);
qed "prefix_antisym";
(** recursion equations **)
Goalw [prefix_def] "[] <= xs";
by (simp_tac (simpset() addsimps [eq_sym_conv]) 1);
qed "Nil_prefix";
AddIffs[Nil_prefix];
Goalw [prefix_def] "(xs <= []) = (xs = [])";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (Simp_tac 1);
qed "prefix_Nil";
Addsimps [prefix_Nil];
Goalw [prefix_def] "(xs <= ys@[y]) = (xs = ys@[y] | xs <= ys)";
by (rtac iffI 1);
by (etac exE 1);
by (rename_tac "zs" 1);
by (res_inst_tac [("xs","zs")] rev_exhaust 1);
by (Asm_full_simp_tac 1);
by (hyp_subst_tac 1);
by (asm_full_simp_tac (simpset() delsimps [append_assoc]
addsimps [append_assoc RS sym])1);
by (etac disjE 1);
by (Asm_simp_tac 1);
by (Clarify_tac 1);
by (Simp_tac 1);
qed "prefix_snoc";
Addsimps [prefix_snoc];
Goalw [prefix_def] "(x#xs <= y#ys) = (x=y & xs<=ys)";
by (Simp_tac 1);
by (Fast_tac 1);
qed"Cons_prefix_Cons";
Addsimps [Cons_prefix_Cons];
Goal "(xs@ys <= xs@zs) = (ys <= zs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "same_prefix_prefix";
Addsimps [same_prefix_prefix];
AddIffs (* (xs@ys <= xs) = (ys <= []) *)
[simplify (simpset()) (read_instantiate [("zs","[]")] same_prefix_prefix)];
Goalw [prefix_def] "xs <= ys ==> xs <= ys@zs";
by (Clarify_tac 1);
by (Simp_tac 1);
qed "prefix_prefix";
Addsimps [prefix_prefix];
Goalw [prefix_def]
"(xs <= y#ys) = (xs=[] | (? zs. xs=y#zs & zs <= ys))";
by (exhaust_tac "xs" 1);
by Auto_tac;
qed "prefix_Cons";
Goal "(xs <= ys@zs) = (xs <= ys | (? us. xs = ys@us & us <= zs))";
by (res_inst_tac [("xs","zs")] rev_induct 1);
by (Simp_tac 1);
by (Blast_tac 1);
by (asm_full_simp_tac (simpset() delsimps [append_assoc]
addsimps [append_assoc RS sym])1);
by (Simp_tac 1);
by (Blast_tac 1);
qed "prefix_append";
Goalw [prefix_def]
"[| xs <= ys; length xs < length ys |] ==> xs @ [ys ! length xs] <= ys";
by (auto_tac(claset(), simpset() addsimps [nth_append]));
by (exhaust_tac "ys" 1);
by Auto_tac;
qed "append_one_prefix";
Goalw [prefix_def] "xs <= ys ==> length xs <= length ys";
by Auto_tac;
qed "prefix_length_le";