Quelle  upair.thy

(*  Title:      ZF/upair.thy
  Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
  Copyright 1993 University of Cambridge
 
 Observe the order of dependence:
  Upair is defined in terms of Replace
  ∪ is defined in terms of Upair and ∪(similarly for Int)
  cons is defined in terms of Upair and Un
  Ordered pairs and descriptions are defined using cons ("set notation")
*)

section‹Unordered Pairs›

theory upair
imports ZF_Base
keywords "print_tcset" :: diag
begin

ML_file ‹Tools/typechk.ML›


subsection‹Unordered Pairs: constant 🍋‹Upair›\›

lemma Upair_iff [simp]: "c ∈ Upair(a,b) ⟷ (c=a | c=b)"
by (unfold Upair_def, blast)

lemma UpairI1: "a ∈ Upair(a,b)"
by simp

lemma UpairI2: "b ∈ Upair(a,b)"
by simp

lemma UpairE: "[a ∈ Upair(b,c); a=b ==> P; a=c ==> P] ==> P"
by (simp, blast)

subsection‹Rules for Binary Union, Defined via 🍋‹Upair›\›

lemma Un_iff [simp]: "c ∈ A ∪ B ⟷ (c ∈ A | c ∈ B)"
apply (simp add: Un_def)
apply (blast intro: UpairI1 UpairI2 elim: UpairE)
done

lemma UnI1: "c ∈ A ==> c ∈ A ∪ B"
by simp

lemma UnI2: "c ∈ B ==> c ∈ A ∪ B"
by simp

declare UnI1 [elim?]  UnI2 [elim?]

lemma UnE [elim!]: "[c ∈ A ∪ B; c ∈ A ==> P; c ∈ B ==> P] ==> P"
by (simp, blast)

(*Stronger version of the rule above*)
lemma UnE': "[c ∈ A ∪ B; c ∈ A ==> P; [c ∈ B; c∉A] ==> P] ==> P"
by (simp, blast)

(*Classical introduction rule: no commitment to A vs B*)
lemma UnCI [intro!]: "(c ∉ B ==> c ∈ A) ==> c ∈ A ∪ B"
by (simp, blast)

subsection‹Rules for Binary Intersection, Defined via 🍋‹Upair›\
lemma Int_iff [simp]: "c ∈ A ∩ B ⟷ (c ∈ A ∧ c ∈ B)"
  unfolding Int_def
apply (blast intro: UpairI1 UpairI2 elim: UpairE)
done

lemma IntI [intro!]: "[c ∈ A; c ∈ B] ==> c ∈ A ∩ B"
by simp

lemma IntD1: "c ∈ A ∩ B ==> c ∈ A"
by simp

lemma IntD2: "c ∈ A ∩ B ==> c ∈ B"
by simp

lemma IntE [elim!]: "[c ∈ A ∩ B; [c ∈ A; c ∈ B] ==> P] ==> P"
by simp


subsection‹Rules for Set Difference, Defined via 🍋‹Upair›\
lemma Diff_iff [simp]: "c ∈ A-B ⟷ (c ∈ A ∧ c∉B)"
by (unfold Diff_def, blast)

lemma DiffI [intro!]: "[c ∈ A; c ∉ B] ==> c ∈ A - B"
by simp

lemma DiffD1: "c ∈ A - B ==> c ∈ A"
by simp

lemma DiffD2: "c ∈ A - B ==> c ∉ B"
by simp

lemma DiffE [elim!]: "[c ∈ A - B; [c ∈ A; c∉B] ==> P] ==> P"
by simp


subsection‹Rules for 🍋‹cons›\›

lemma cons_iff [simp]: "a ∈ cons(b,A) ⟷ (a=b | a ∈ A)"
  unfolding cons_def
apply (blast intro: UpairI1 UpairI2 elim: UpairE)
done

(*risky as a typechecking rule, but solves otherwise unconstrained goals of
the form x \<in> ?A*)
lemma consI1 [simp,TC]: "a ∈ cons(a,B)"
by simp


lemma consI2: "a ∈ B ==> a ∈ cons(b,B)"
by simp

lemma consE [elim!]: "[a ∈ cons(b,A); a=b ==> P; a ∈ A ==> P] ==> P"
by (simp, blast)

(*Stronger version of the rule above*)
lemma consE':
    "[a ∈ cons(b,A); a=b ==> P; [a ∈ A; a≠b] ==> P] ==> P"
by (simp, blast)

(*Classical introduction rule*)
lemma consCI [intro!]: "(a∉B ==> a=b) ==> a ∈ cons(b,B)"
by (simp, blast)

lemma cons_not_0 [simp]: "cons(a,B) ≠ 0"
by (blast elim: equalityE)

lemmas cons_neq_0 = cons_not_0 [THEN notE]

declare cons_not_0 [THEN not_sym, simp]


subsection‹Singletons›

lemma singleton_iff: "a ∈ {b} ⟷ a=b"
by simp

lemma singletonI [intro!]: "a ∈ {a}"
by (rule consI1)

lemmas singletonE = singleton_iff [THEN iffD1, elim_format, elim!]


subsection‹Descriptions›

lemma the_equality [intro]:
    "[P(a); ∧x. P(x) ==> x=a] ==> (THE x. P(x)) = a"
  unfolding the_def
apply (fast dest: subst)
done

(* Only use this if you already know \<exists>!x. P(x) *)
lemma the_equality2: "[∃!x. P(x); P(a)] ==> (THE x. P(x)) = a"
by blast

lemma theI: "∃!x. P(x) ==> P(THE x. P(x))"
apply (erule ex1E)
apply (subst the_equality)
apply (blast+)
done

(*No congruence rule is necessary: if @{term"\<forall>y.P(y)\<longleftrightarrow>Q(y)"} then
  @{term "THE x.P(x)"}  rewrites to @{term "THE x.Q(x)"} *)

(*If it's "undefined", it's zero!*)
lemma the_0: "¬ (∃!x. P(x)) ==> (THE x. P(x))=0"
  unfolding the_def
apply (blast elim!: ReplaceE)
done

(*Easier to apply than theI: conclusion has only one occurrence of P*)
lemma theI2:
    assumes p1: "¬ Q(0) ==> ∃!x. P(x)"
        and p2: "∧x. P(x) ==> Q(x)"
    shows "Q(THE x. P(x))"
apply (rule classical)
apply (rule p2)
apply (rule theI)
apply (rule classical)
apply (rule p1)
apply (erule the_0 [THEN subst], assumption)
done

lemma the_eq_trivial [simp]: "(THE x. x = a) = a"
by blast

lemma the_eq_trivial2 [simp]: "(THE x. a = x) = a"
by blast


subsection‹Conditional Terms: ‹if-then-else›\
lemma if_true [simp]: "(if True then a else b) = a"
by (unfold if_def, blast)

lemma if_false [simp]: "(if False then a else b) = b"
by (unfold if_def, blast)

(*Never use with case splitting, or if P is known to be true or false*)
lemma if_cong:
    "[P⟷Q; Q ==> a=c; ¬Q ==> b=d]
     ==> (if P then a else b) = (if Q then c else d)"
by (simp add: if_def cong add: conj_cong)

(*Prevents simplification of x and y \<in> faster and allows the execution
  of functional programs. NOW THE DEFAULT.*)
lemma if_weak_cong: "P⟷Q ==> (if P then x else y) = (if Q then x else y)"
by simp

(*Not needed for rewriting, since P would rewrite to True anyway*)
lemma if_P: "P ==> (if P then a else b) = a"
by (unfold if_def, blast)

(*Not needed for rewriting, since P would rewrite to False anyway*)
lemma if_not_P: "¬P ==> (if P then a else b) = b"
by (unfold if_def, blast)

lemma split_if [split]:
     "P(if Q then x else y) ⟷ ((Q ⟶ P(x)) ∧ (¬Q ⟶ P(y)))"
by (case_tac Q, simp_all)

(** Rewrite rules for boolean case-splitting: faster than split_if [split]
**)

lemmas split_if_eq1 = split_if [of "λx. x = b"] for b
lemmas split_if_eq2 = split_if [of "λx. a = x"] for a

lemmas split_if_mem1 = split_if [of "λx. x ∈ b"] for b
lemmas split_if_mem2 = split_if [of "λx. a ∈ x"] for a

lemmas split_ifs = split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2

(*Logically equivalent to split_if_mem2*)
lemma if_iff: "a: (if P then x else y) ⟷ P ∧ a ∈ x | ¬P ∧ a ∈ y"
by simp

lemma if_type [TC]:
    "[P ==> a ∈ A; ¬P ==> b ∈ A] ==> (if P then a else b): A"
by simp

(** Splitting IFs in the assumptions **)

lemma split_if_asm: "P(if Q then x else y) ⟷ (¬((Q ∧ ¬P(x)) | (¬Q ∧ ¬P(y))))"
by simp

lemmas if_splits = split_if split_if_asm


subsection‹Consequences of Foundation›

(*was called mem_anti_sym*)
lemma mem_asym: "[a ∈ b; ¬P ==> b ∈ a] ==> P"
apply (rule classical)
apply (rule_tac A1 = "{a,b}" in foundation [THEN disjE])
apply (blast elim!: equalityE)+
done

(*was called mem_anti_refl*)
lemma mem_irrefl: "a ∈ a ==> P"
by (blast intro: mem_asym)

(*mem_irrefl should NOT be added to default databases:
      it would be tried on most goals, making proofs slower!*)

lemma mem_not_refl: "a ∉ a"
apply (rule notI)
apply (erule mem_irrefl)
done

(*Good for proving inequalities by rewriting*)
lemma mem_imp_not_eq: "a ∈ A ==> a ≠ A"
by (blast elim!: mem_irrefl)

lemma eq_imp_not_mem: "a=A ==> a ∉ A"
by (blast intro: elim: mem_irrefl)

subsection‹Rules for Successor›

lemma succ_iff: "i ∈ succ(j) ⟷ i=j | i ∈ j"
by (unfold succ_def, blast)

lemma succI1 [simp]: "i ∈ succ(i)"
by (simp add: succ_iff)

lemma succI2: "i ∈ j ==> i ∈ succ(j)"
by (simp add: succ_iff)

lemma succE [elim!]:
    "[i ∈ succ(j); i=j ==> P; i ∈ j ==> P] ==> P"
apply (simp add: succ_iff, blast)
done

(*Classical introduction rule*)
lemma succCI [intro!]: "(i∉j ==> i=j) ==> i ∈ succ(j)"
by (simp add: succ_iff, blast)

lemma succ_not_0 [simp]: "succ(n) ≠ 0"
by (blast elim!: equalityE)

lemmas succ_neq_0 = succ_not_0 [THEN notE, elim!]

declare succ_not_0 [THEN not_sym, simp]
declare sym [THEN succ_neq_0, elim!]

(* @{term"succ(c) \<subseteq> B \<Longrightarrow> c \<in> B"} *)
lemmas succ_subsetD = succI1 [THEN [2] subsetD]

(* @{term"succ(b) \<noteq> b"} *)
lemmas succ_neq_self = succI1 [THEN mem_imp_not_eq, THEN not_sym]

lemma succ_inject_iff [simp]: "succ(m) = succ(n) ⟷ m=n"
by (blast elim: mem_asym elim!: equalityE)

lemmas succ_inject = succ_inject_iff [THEN iffD1, dest!]


subsection‹Miniscoping of the Bounded Universal Quantifier›

lemma ball_simps1:
     "(∀x∈A. P(x) ∧ Q) ⟷ (∀x∈A. P(x)) ∧ (A=0 | Q)"
     "(∀x∈A. P(x) | Q) ⟷ ((∀x∈A. P(x)) | Q)"
     "(∀x∈A. P(x) ⟶ Q) ⟷ ((∃x∈A. P(x)) ⟶ Q)"
     "(¬(∀x∈A. P(x))) ⟷ (∃x∈A. ¬P(x))"
     "(∀x∈0.P(x)) ⟷ True"
     "(∀x∈succ(i).P(x)) ⟷ P(i) ∧ (∀x∈i. P(x))"
     "(∀x∈cons(a,B).P(x)) ⟷ P(a) ∧ (∀x∈B. P(x))"
     "(∀x∈RepFun(A,f). P(x)) ⟷ (∀y∈A. P(f(y)))"
     "(∀x∈∪(A).P(x)) ⟷ (∀y∈A. ∀x∈y. P(x))"
by blast+

lemma ball_simps2:
     "(∀x∈A. P ∧ Q(x)) ⟷ (A=0 | P) ∧ (∀x∈A. Q(x))"
     "(∀x∈A. P | Q(x)) ⟷ (P | (∀x∈A. Q(x)))"
     "(∀x∈A. P ⟶ Q(x)) ⟷ (P ⟶ (∀x∈A. Q(x)))"
by blast+

lemma ball_simps3:
     "(∀x∈Collect(A,Q).P(x)) ⟷ (∀x∈A. Q(x) ⟶ P(x))"
by blast+

lemmas ball_simps [simp] = ball_simps1 ball_simps2 ball_simps3

lemma ball_conj_distrib:
    "(∀x∈A. P(x) ∧ Q(x)) ⟷ ((∀x∈A. P(x)) ∧ (∀x∈A. Q(x)))"
by blast


subsection‹Miniscoping of the Bounded Existential Quantifier›

lemma bex_simps1:
     "(∃x∈A. P(x) ∧ Q) ⟷ ((∃x∈A. P(x)) ∧ Q)"
     "(∃x∈A. P(x) | Q) ⟷ (∃x∈A. P(x)) | (A≠0 ∧ Q)"
     "(∃x∈A. P(x) ⟶ Q) ⟷ ((∀x∈A. P(x)) ⟶ (A≠0 ∧ Q))"
     "(∃x∈0.P(x)) ⟷ False"
     "(∃x∈succ(i).P(x)) ⟷ P(i) | (∃x∈i. P(x))"
     "(∃x∈cons(a,B).P(x)) ⟷ P(a) | (∃x∈B. P(x))"
     "(∃x∈RepFun(A,f). P(x)) ⟷ (∃y∈A. P(f(y)))"
     "(∃x∈∪(A).P(x)) ⟷ (∃y∈A. ∃x∈y. P(x))"
     "(¬(∃x∈A. P(x))) ⟷ (∀x∈A. ¬P(x))"
by blast+

lemma bex_simps2:
     "(∃x∈A. P ∧ Q(x)) ⟷ (P ∧ (∃x∈A. Q(x)))"
     "(∃x∈A. P | Q(x)) ⟷ (A≠0 ∧ P) | (∃x∈A. Q(x))"
     "(∃x∈A. P ⟶ Q(x)) ⟷ ((A=0 | P) ⟶ (∃x∈A. Q(x)))"
by blast+

lemma bex_simps3:
     "(∃x∈Collect(A,Q).P(x)) ⟷ (∃x∈A. Q(x) ∧ P(x))"
by blast

lemmas bex_simps [simp] = bex_simps1 bex_simps2 bex_simps3

lemma bex_disj_distrib:
    "(∃x∈A. P(x) | Q(x)) ⟷ ((∃x∈A. P(x)) | (∃x∈A. Q(x)))"
by blast


(** One-point rule for bounded quantifiers: see HOL/Set.ML **)

lemma bex_triv_one_point1 [simp]: "(∃x∈A. x=a) ⟷ (a ∈ A)"
by blast

lemma bex_triv_one_point2 [simp]: "(∃x∈A. a=x) ⟷ (a ∈ A)"
by blast

lemma bex_one_point1 [simp]: "(∃x∈A. x=a ∧ P(x)) ⟷ (a ∈ A ∧ P(a))"
by blast

lemma bex_one_point2 [simp]: "(∃x∈A. a=x ∧ P(x)) ⟷ (a ∈ A ∧ P(a))"
by blast

lemma ball_one_point1 [simp]: "(∀x∈A. x=a ⟶ P(x)) ⟷ (a ∈ A ⟶ P(a))"
by blast

lemma ball_one_point2 [simp]: "(∀x∈A. a=x ⟶ P(x)) ⟷ (a ∈ A ⟶ P(a))"
by blast


subsection‹Miniscoping of the Replacement Operator›

text‹These cover both 🍋‹Replace› and 🍋‹Collect›\<close>
lemma Rep_simps [simp]:
     "{x. y ∈ 0, R(x,y)} = 0"
     "{x ∈ 0. P(x)} = 0"
     "{x ∈ A. Q} = (if Q then A else 0)"
     "RepFun(0,f) = 0"
     "RepFun(succ(i),f) = cons(f(i), RepFun(i,f))"
     "RepFun(cons(a,B),f) = cons(f(a), RepFun(B,f))"
by (simp_all, blast+)


subsection‹Miniscoping of Unions›

lemma UN_simps1:
     "(∪x∈C. cons(a, B(x))) = (if C=0 then 0 else cons(a, ∪x∈C. B(x)))"
     "(∪x∈C. A(x) ∪ B') = (if C=0 then 0 else (∪x∈C. A(x)) ∪ B')"
     "(∪x∈C. A' ∪ B(x)) = (if C=0 then 0 else A' ∪ (∪x∈C. B(x)))"
     "(∪x∈C. A(x) ∩ B') = ((∪x∈C. A(x)) ∩ B')"
     "(∪x∈C. A' ∩ B(x)) = (A' ∩ (∪x∈C. B(x)))"
     "(∪x∈C. A(x) - B') = ((∪x∈C. A(x)) - B')"
     "(∪x∈C. A' - B(x)) = (if C=0 then 0 else A' - (∩x∈C. B(x)))"
apply (simp_all add: Inter_def)
apply (blast intro!: equalityI )+
done

lemma UN_simps2:
      "(∪x∈∪(A). B(x)) = (∪y∈A. ∪x∈y. B(x))"
      "(∪z∈(∪x∈A. B(x)). C(z)) = (∪x∈A. ∪z∈B(x). C(z))"
      "(∪x∈RepFun(A,f). B(x)) = (∪a∈A. B(f(a)))"
by blast+

lemmas UN_simps [simp] = UN_simps1 UN_simps2

text‹Opposite of miniscoping: pull the operator out›

lemma UN_extend_simps1:
     "(∪x∈C. A(x)) ∪ B = (if C=0 then B else (∪x∈C. A(x) ∪ B))"
     "((∪x∈C. A(x)) ∩ B) = (∪x∈C. A(x) ∩ B)"
     "((∪x∈C. A(x)) - B) = (∪x∈C. A(x) - B)"
apply simp_all
apply blast+
done

lemma UN_extend_simps2:
     "cons(a, ∪x∈C. B(x)) = (if C=0 then {a} else (∪x∈C. cons(a, B(x))))"
     "A ∪ (∪x∈C. B(x)) = (if C=0 then A else (∪x∈C. A ∪ B(x)))"
     "(A ∩ (∪x∈C. B(x))) = (∪x∈C. A ∩ B(x))"
     "A - (∩x∈C. B(x)) = (if C=0 then A else (∪x∈C. A - B(x)))"
     "(∪y∈A. ∪x∈y. B(x)) = (∪x∈∪(A). B(x))"
     "(∪a∈A. B(f(a))) = (∪x∈RepFun(A,f). B(x))"
apply (simp_all add: Inter_def)
apply (blast intro!: equalityI)+
done

lemma UN_UN_extend:
     "(∪x∈A. ∪z∈B(x). C(z)) = (∪z∈(∪x∈A. B(x)). C(z))"
by blast

lemmas UN_extend_simps = UN_extend_simps1 UN_extend_simps2 UN_UN_extend


subsection‹Miniscoping of Intersections›

lemma INT_simps1:
     "(∩x∈C. A(x) ∩ B) = (∩x∈C. A(x)) ∩ B"
     "(∩x∈C. A(x) - B) = (∩x∈C. A(x)) - B"
     "(∩x∈C. A(x) ∪ B) = (if C=0 then 0 else (∩x∈C. A(x)) ∪ B)"
by (simp_all add: Inter_def, blast+)

lemma INT_simps2:
     "(∩x∈C. A ∩ B(x)) = A ∩ (∩x∈C. B(x))"
     "(∩x∈C. A - B(x)) = (if C=0 then 0 else A - (∪x∈C. B(x)))"
     "(∩x∈C. cons(a, B(x))) = (if C=0 then 0 else cons(a, ∩x∈C. B(x)))"
     "(∩x∈C. A ∪ B(x)) = (if C=0 then 0 else A ∪ (∩x∈C. B(x)))"
apply (simp_all add: Inter_def)
apply (blast intro!: equalityI)+
done

lemmas INT_simps [simp] = INT_simps1 INT_simps2

text‹Opposite of miniscoping: pull the operator out›


lemma INT_extend_simps1:
     "(∩x∈C. A(x)) ∩ B = (∩x∈C. A(x) ∩ B)"
     "(∩x∈C. A(x)) - B = (∩x∈C. A(x) - B)"
     "(∩x∈C. A(x)) ∪ B = (if C=0 then B else (∩x∈C. A(x) ∪ B))"
apply (simp_all add: Inter_def, blast+)
done

lemma INT_extend_simps2:
     "A ∩ (∩x∈C. B(x)) = (∩x∈C. A ∩ B(x))"
     "A - (∪x∈C. B(x)) = (if C=0 then A else (∩x∈C. A - B(x)))"
     "cons(a, ∩x∈C. B(x)) = (if C=0 then {a} else (∩x∈C. cons(a, B(x))))"
     "A ∪ (∩x∈C. B(x)) = (if C=0 then A else (∩x∈C. A ∪ B(x)))"
apply (simp_all add: Inter_def)
apply (blast intro!: equalityI)+
done

lemmas INT_extend_simps = INT_extend_simps1 INT_extend_simps2


subsection‹Other simprules›


(*** Miniscoping: pushing in big Unions, Intersections, quantifiers, etc. ***)

lemma misc_simps [simp]:
     "0 ∪ A = A"
     "A ∪ 0 = A"
     "0 ∩ A = 0"
     "A ∩ 0 = 0"
     "0 - A = 0"
     "A - 0 = A"
     "∪(0) = 0"
     "∪(cons(b,A)) = b ∪ ∪(A)"
     "∩({b}) = b"
by blast+

end