(* Title: HOL/Bali/Basis.thy
Author: David von Oheimb
*)
subsection \<open>Definitions extending HOL as logical basis of Bali\<close>
theory Basis
imports Main
begin
subsubsection "misc"
ML \<open>fun strip_tac ctxt i = REPEAT (resolve_tac ctxt [impI, allI] i)\<close>
declare if_split_asm [split] option.split [split] option.split_asm [split]
setup \<open>map_theory_simpset (fn ctxt => ctxt addloop ("split_all_tac", split_all_tac))\<close>
declare if_weak_cong [cong del] option.case_cong_weak [cong del]
declare length_Suc_conv [iff]
lemma Collect_split_eq: "{p. P (case_prod f p)} = {(a,b). P (f a b)}"
by auto
lemma subset_insertD: "A \ insert x B \ A \ B \ x \ A \ (\B'. A = insert x B' \ B' \ B)"
apply (case_tac "x \ A")
apply (rule disjI2)
apply (rule_tac x = "A - {x}" in exI)
apply fast+
done
abbreviation nat3 :: nat ("3") where "3 \ Suc 2"
abbreviation nat4 :: nat ("4") where "4 \ Suc 3"
(* irrefl_tranclI in Transitive_Closure.thy is more general *)
lemma irrefl_tranclI': "r\ \ r\<^sup>+ = {} \ \x. (x, x) \ r\<^sup>+"
by (blast elim: tranclE dest: trancl_into_rtrancl)
lemma trancl_rtrancl_trancl: "\(x, y) \ r\<^sup>+; (y, z) \ r\<^sup>*\ \ (x, z) \ r\<^sup>+"
by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
lemma rtrancl_into_trancl3: "\(a, b) \ r\<^sup>*; a \ b\ \ (a, b) \ r\<^sup>+"
apply (drule rtranclD)
apply auto
done
lemma rtrancl_into_rtrancl2: "\(a, b) \ r; (b, c) \ r\<^sup>*\ \ (a, c) \ r\<^sup>*"
by (auto intro: rtrancl_trans)
lemma triangle_lemma:
assumes unique: "\a b c. \(a,b)\r; (a,c)\r\ \ b = c"
and ax: "(a,x)\r\<^sup>*" and ay: "(a,y)\r\<^sup>*"
shows "(x,y)\r\<^sup>* \ (y,x)\r\<^sup>*"
using ax ay
proof (induct rule: converse_rtrancl_induct)
assume "(x,y)\r\<^sup>*"
then show ?thesis by blast
next
fix a v
assume a_v_r: "(a, v) \ r"
and v_x_rt: "(v, x) \ r\<^sup>*"
and a_y_rt: "(a, y) \ r\<^sup>*"
and hyp: "(v, y) \ r\<^sup>* \ (x, y) \ r\<^sup>* \ (y, x) \ r\<^sup>*"
from a_y_rt show "(x, y) \ r\<^sup>* \ (y, x) \ r\<^sup>*"
proof (cases rule: converse_rtranclE)
assume "a = y"
with a_v_r v_x_rt have "(y,x) \ r\<^sup>*"
by (auto intro: rtrancl_trans)
then show ?thesis by blast
next
fix w
assume a_w_r: "(a, w) \ r"
and w_y_rt: "(w, y) \ r\<^sup>*"
from a_v_r a_w_r unique have "v=w" by auto
with w_y_rt hyp show ?thesis by blast
qed
qed
lemma rtrancl_cases:
assumes "(a,b)\r\<^sup>*"
obtains (Refl) "a = b"
| (Trancl) "(a,b)\r\<^sup>+"
apply (rule rtranclE [OF assms])
apply (auto dest: rtrancl_into_trancl1)
done
lemma Ball_weaken: "\Ball s P; \ x. P x\Q x\\Ball s Q"
by auto
lemma finite_SetCompr2:
"finite {f y x |x y. P y}" if "finite (Collect P)"
"\y. P y \ finite (range (f y))"
proof -
have "{f y x |x y. P y} = (\y\Collect P. range (f y))"
by auto
with that show ?thesis by simp
qed
lemma list_all2_trans: "\a b c. P1 a b \ P2 b c \ P3 a c \
\<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
apply (induct_tac xs1)
apply simp
apply (rule allI)
apply (induct_tac xs2)
apply simp
apply (rule allI)
apply (induct_tac xs3)
apply auto
done
subsubsection "pairs"
lemma surjective_pairing5:
"p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
snd (snd (snd (snd p))))"
by auto
lemma fst_splitE [elim!]:
assumes "fst s' = x'"
obtains x s where "s' = (x,s)" and "x = x'"
using assms by (cases s') auto
lemma fst_in_set_lemma: "(x, y) \ set l \ x \ fst ` set l"
by (induct l) auto
subsubsection "quantifiers"
lemma All_Ex_refl_eq2 [simp]: "(\x. (\b. x = f b \ Q b) \ P x) = (\b. Q b \ P (f b))"
by auto
lemma ex_ex_miniscope1 [simp]: "(\w v. P w v \ Q v) = (\v. (\w. P w v) \ Q v)"
by auto
lemma ex_miniscope2 [simp]: "(\v. P v \ Q \ R v) = (Q \ (\v. P v \ R v))"
by auto
lemma ex_reorder31: "(\z x y. P x y z) = (\x y z. P x y z)"
by auto
lemma All_Ex_refl_eq1 [simp]: "(\x. (\b. x = f b) \ P x) = (\b. P (f b))"
by auto
subsubsection "sums"
notation case_sum (infixr "'(+')" 80)
primrec the_Inl :: "'a + 'b \ 'a"
where "the_Inl (Inl a) = a"
primrec the_Inr :: "'a + 'b \ 'b"
where "the_Inr (Inr b) = b"
datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
primrec the_In1 :: "('a, 'b, 'c) sum3 \ 'a"
where "the_In1 (In1 a) = a"
primrec the_In2 :: "('a, 'b, 'c) sum3 \ 'b"
where "the_In2 (In2 b) = b"
primrec the_In3 :: "('a, 'b, 'c) sum3 \ 'c"
where "the_In3 (In3 c) = c"
abbreviation In1l :: "'al \ ('al + 'ar, 'b, 'c) sum3"
where "In1l e \ In1 (Inl e)"
abbreviation In1r :: "'ar \ ('al + 'ar, 'b, 'c) sum3"
where "In1r c \ In1 (Inr c)"
abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \ 'al"
where "the_In1l \ the_Inl \ the_In1"
abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \ 'ar"
where "the_In1r \ the_Inr \ the_In1"
ML \<open>
fun sum3_instantiate ctxt thm =
map (fn s =>
simplify (ctxt delsimps @{thms not_None_eq})
(Rule_Insts.read_instantiate ctxt [((("t", 0), Position.none), "In" ^ s ^ " x")] ["x"] thm))
["1l","2","3","1r"]
\<close>
(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
subsubsection "quantifiers for option type"
syntax
"_Oall" :: "[pttrn, 'a option, bool] \ bool" ("(3! _:_:/ _)" [0,0,10] 10)
"_Oex" :: "[pttrn, 'a option, bool] \ bool" ("(3? _:_:/ _)" [0,0,10] 10)
syntax (symbols)
"_Oall" :: "[pttrn, 'a option, bool] \ bool" ("(3\_\_:/ _)" [0,0,10] 10)
"_Oex" :: "[pttrn, 'a option, bool] \ bool" ("(3\_\_:/ _)" [0,0,10] 10)
translations
"\x\A: P" \ "\x\CONST set_option A. P"
"\x\A: P" \ "\x\CONST set_option A. P"
subsubsection "Special map update"
text\<open>Deemed too special for theory Map.\<close>
definition chg_map :: "('b \ 'b) \ 'a \ ('a \ 'b) \ ('a \ 'b)"
where "chg_map f a m = (case m a of None \ m | Some b \ m(a\f b))"
lemma chg_map_new[simp]: "m a = None \ chg_map f a m = m"
unfolding chg_map_def by auto
lemma chg_map_upd[simp]: "m a = Some b \ chg_map f a m = m(a\f b)"
unfolding chg_map_def by auto
lemma chg_map_other [simp]: "a \ b \ chg_map f a m b = m b"
by (auto simp: chg_map_def)
subsubsection "unique association lists"
definition unique :: "('a \ 'b) list \ bool"
where "unique = distinct \ map fst"
lemma uniqueD: "unique l \ (x, y) \ set l \ (x', y') \ set l \ x = x' \ y = y'"
unfolding unique_def o_def
by (induct l) (auto dest: fst_in_set_lemma)
lemma unique_Nil [simp]: "unique []"
by (simp add: unique_def)
lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \ (\y. (x,y) \ set l))"
by (auto simp: unique_def dest: fst_in_set_lemma)
lemma unique_ConsD: "unique (x#xs) \ unique xs"
by (simp add: unique_def)
lemma unique_single [simp]: "\p. unique [p]"
by simp
lemma unique_append [rule_format (no_asm)]: "unique l' \ unique l \
(\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"
by (induct l) (auto dest: fst_in_set_lemma)
lemma unique_map_inj: "unique l \ inj f \ unique (map (\(k,x). (f k, g k x)) l)"
by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)
lemma map_of_SomeI: "unique l \ (k, x) \ set l \ map_of l k = Some x"
by (induct l) auto
subsubsection "list patterns"
definition lsplit :: "[['a, 'a list] \ 'b, 'a list] \ 'b"
where "lsplit = (\f l. f (hd l) (tl l))"
text \<open>list patterns -- extends pre-defined type "pttrn" used in abstractions\<close>
syntax
"_lpttrn" :: "[pttrn, pttrn] \ pttrn" ("_#/_" [901,900] 900)
translations
"\y # x # xs. b" \ "CONST lsplit (\y x # xs. b)"
"\x # xs. b" \ "CONST lsplit (\x xs. b)"
lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
by (simp add: lsplit_def)
lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
by (simp add: lsplit_def)
end
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