(* Title: isabelle/Bali/Basis.thy
ID: $Id$
Author: David von Oheimb
Copyright 1997 Technische Universitaet Muenchen
*)
header {* Definitions extending HOL as logical basis of Bali *}
theory Basis = Main:
ML_setup {*
Unify.search_bound := 40;
Unify.trace_bound := 40;
quick_and_dirty:=true;
Pretty.setmargin 77;
goals_limit:=2;
*}
(*print_depth 100;*)
(*Syntax.ambiguity_level := 1;*)
section "misc"
declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
(* ###TO HOL/???.ML?? *)
ML {*
fun make_simproc name pat pred thm = Simplifier.mk_simproc name
[Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.typeT)]
(K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))
*}
declare split_if_asm [split] option.split [split] option.split_asm [split]
ML {*
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
*}
declare if_weak_cong [cong del] option.weak_case_cong [cong del]
declare length_Suc_conv [iff];
(*###to be phased out *)
ML {*
bind_thm ("make_imp", rearrange_prems [1,0] mp)
*}
lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
apply auto
done
lemma subset_insertD:
"A <= insert x B ==> A <= B & x ~: A | (EX 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
syntax
"3" :: nat ("3")
"4" :: nat ("4")
translations
"3" == "Suc 2"
"4" == "Suc 3"
(*unused*)
lemma range_bool_domain: "range f = {f True, f False}"
apply auto
apply (case_tac "xa")
apply auto
done
(* context (theory "Transitive_Closure") *)
lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
apply (rule allI)
apply (erule irrefl_tranclI)
done
lemma trancl_rtrancl_trancl:
"\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
lemma rtrancl_into_trancl3:
"\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"
apply (drule rtranclD)
apply auto
done
lemma rtrancl_into_rtrancl2:
"\<lbrakk> (a, b) \<in> r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in> r^*"
by (auto intro: r_into_rtrancl rtrancl_trans)
lemma triangle_lemma:
"\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk>
\<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
proof -
note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
note converse_rtranclE = converse_rtranclE [consumes 1]
assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
assume "(a,x)\<in>r\<^sup>*"
then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
proof (induct rule: converse_rtrancl_induct)
assume "(x,y)\<in>r\<^sup>*"
then show ?thesis
by blast
next
fix a v
assume a_v_r: "(a, v) \<in> r" and
v_x_rt: "(v, x) \<in> r\<^sup>*" and
a_y_rt: "(a, y) \<in> r\<^sup>*" and
hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
from a_y_rt
show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
proof (cases rule: converse_rtranclE)
assume "a=y"
with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
by (auto intro: r_into_rtrancl rtrancl_trans)
then show ?thesis
by blast
next
fix w
assume a_w_r: "(a, w) \<in> r" and
w_y_rt: "(w, y) \<in> 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
qed
lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
"\<lbrakk>(a,b)\<in>r\<^sup>*; a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (erule rtranclE)
apply (auto dest: rtrancl_into_trancl1)
done
(* ### To Transitive_Closure *)
theorems converse_rtrancl_induct
= converse_rtrancl_induct [consumes 1,case_names Id Step]
theorems converse_trancl_induct
= converse_trancl_induct [consumes 1,case_names Single Step]
(* context (theory "Set") *)
lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
by auto
(* context (theory "Finite") *)
lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>
finite {f y x |x y. P y}"
apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
prefer 2 apply fast
apply (erule ssubst)
apply (erule finite_UN_I)
apply fast
done
(* ### TO theory "List" *)
lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
\<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
section "pairs"
lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
snd (snd (snd (snd p))))"
apply auto
done
lemma fst_splitE [elim!]:
"[| fst s' = x'; !!x s. [| s' = (x,s); x = x' |] ==> Q |] ==> Q"
apply (cut_tac p = "s'" in surjective_pairing)
apply auto
done
lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
apply (induct_tac "l")
apply auto
done
section "quantifiers"
(*###to be phased out *)
ML {*
fun noAll_simpset () = simpset() setmksimps
mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
*}
lemma All_Ex_refl_eq2 [simp]:
"(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
apply auto
done
lemma ex_ex_miniscope1 [simp]:
"(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
apply auto
done
lemma ex_miniscope2 [simp]:
"(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))"
apply auto
done
lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
apply auto
done
lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
apply auto
done
section "sums"
hide const In0 In1
syntax
fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
translations
"fun_sum" == "sum_case"
consts the_Inl :: "'a + 'b \<Rightarrow> 'a"
the_Inr :: "'a + 'b \<Rightarrow> 'b"
primrec "the_Inl (Inl a) = a"
primrec "the_Inr (Inr b) = b"
datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
consts the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
primrec "the_In1 (In1 a) = a"
primrec "the_In2 (In2 b) = b"
primrec "the_In3 (In3 c) = c"
syntax
In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
translations
"In1l e" == "In1 (Inl e)"
"In1r c" == "In1 (Inr c)"
ML {*
fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
(read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
*}
(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
translations
"option"<= (type) "Option.option"
"list" <= (type) "List.list"
"sum3" <= (type) "Basis.sum3"
section "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\<forall>_\<in>_:/ _)" [0,0,10] 10)
Oex :: "[pttrn, 'a option, bool] => bool" ("(3\<exists>_\<in>_:/ _)" [0,0,10] 10)
translations
"! x:A: P" == "! x:o2s A. P"
"? x:A: P" == "? x:o2s A. P"
section "unique association lists"
constdefs
unique :: "('a \<times> 'b) list \<Rightarrow> bool"
"unique \<equiv> nodups \<circ> map fst"
lemma uniqueD [rule_format (no_asm)]:
"unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'--> y=y'))"
apply (unfold unique_def o_def)
apply (induct_tac "l")
apply (auto dest: fst_in_set_lemma)
done
lemma unique_Nil [simp]: "unique []"
apply (unfold unique_def)
apply (simp (no_asm))
done
lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
apply (unfold unique_def)
apply (auto dest: fst_in_set_lemma)
done
lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
lemma unique_single [simp]: "!!p. unique [p]"
apply auto
done
lemma unique_ConsD: "unique (x#xs) ==> unique xs"
apply (simp add: unique_def)
done
lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->
(!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
apply (induct_tac "l")
apply (auto dest: fst_in_set_lemma)
done
lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
apply (induct_tac "l")
apply (auto dest: fst_in_set_lemma simp add: inj_eq)
done
lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
apply (induct_tac "l")
apply auto
done
section "list patterns"
consts
lsplit :: "[['a, 'a list] => 'b, 'a list] => 'b"
defs
lsplit_def: "lsplit == %f l. f (hd l) (tl l)"
(* list patterns -- extends pre-defined type "pttrn" used in abstractions *)
syntax
"_lpttrn" :: "[pttrn,pttrn] => pttrn" ("_#/_" [901,900] 900)
translations
"%y#x#xs. b" == "lsplit (%y x#xs. b)"
"%x#xs . b" == "lsplit (%x xs . b)"
lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
apply (unfold lsplit_def)
apply (simp (no_asm))
done
lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
apply (unfold lsplit_def)
apply simp
done
section "dummy pattern for quantifiers, let, etc."
syntax
"@dummy_pat" :: pttrn ("'_")
parse_translation {*
let fun dummy_pat_tr [] = Free ("_",dummyT)
| dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
in [("@dummy_pat", dummy_pat_tr)]
end
*}
end