Theory Basis

```(*  Title:      HOL/Bali/Basis.thy
Author:     David von Oheimb
*)
subsection ‹Definitions extending HOL as logical basis of Bali›

theory Basis
imports Main
begin

subsubsection "misc"

ML ‹fun strip_tac ctxt i = REPEAT (resolve_tac ctxt [impI, allI] i)›

declare if_split_asm  [split] option.split [split] option.split_asm [split]
setup ‹map_theory_simpset (fn ctxt => ctxt addloop ("split_all_tac", split_all_tac))›
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⇧+ = {} ⟹ ∀x. (x, x) ∉ r⇧+"
by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma trancl_rtrancl_trancl: "⟦(x, y) ∈ r⇧+; (y, z) ∈ r⇧*⟧ ⟹ (x, z) ∈ r⇧+"
by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)

lemma rtrancl_into_trancl3: "⟦(a, b) ∈ r⇧*; a ≠ b⟧ ⟹ (a, b) ∈ r⇧+"
apply (drule rtranclD)
apply auto
done

lemma rtrancl_into_rtrancl2: "⟦(a, b) ∈  r; (b, c) ∈ r⇧*⟧ ⟹ (a, c) ∈ r⇧*"
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⇧*" and ay: "(a,y)∈r⇧*"
shows "(x,y)∈r⇧* ∨ (y,x)∈r⇧*"
using ax ay
proof (induct rule: converse_rtrancl_induct)
assume "(x,y)∈r⇧*"
then show ?thesis by blast
next
fix a v
assume a_v_r: "(a, v) ∈ r"
and v_x_rt: "(v, x) ∈ r⇧*"
and a_y_rt: "(a, y) ∈ r⇧*"
and hyp: "(v, y) ∈ r⇧* ⟹ (x, y) ∈ r⇧* ∨ (y, x) ∈ r⇧*"
from a_y_rt show "(x, y) ∈ r⇧* ∨ (y, x) ∈ r⇧*"
proof (cases rule: converse_rtranclE)
assume "a = y"
with a_v_r v_x_rt have "(y,x) ∈ r⇧*"
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⇧*"
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⇧*"
obtains (Refl) "a = b"
| (Trancl) "(a,b)∈r⇧+"
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 ⟹
∀xs2 xs3. list_all2 P1 xs1 xs2 ⟶ list_all2 P2 xs2 xs3 ⟶ 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 ‹
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"]
›
(* 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‹Deemed too special for theory Map.›

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 []"

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"

lemma unique_single [simp]: "⋀p. unique [p]"
by simp

lemma unique_append [rule_format (no_asm)]: "unique l' ⟹ unique l ⟹
(∀(x,y)∈set l. ∀(x',y')∈set l'. x' ≠ x) ⟶ unique (l @ l')"
by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_map_inj: "unique l ⟹ inj f ⟹ unique (map (λ(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 = (λf l. f (hd l) (tl l))"

text ‹list patterns -- extends pre-defined type "pttrn" used in abstractions›
syntax
"_lpttrn" :: "[pttrn, pttrn] ⇒ pttrn"    ("_#/_" [901,900] 900)
translations
"λy # x # xs. b" ⇌ "CONST lsplit (λy x # xs. b)"
"λx # xs. b" ⇌ "CONST lsplit (λx xs. b)"

lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"