# Theory pair

```(*  Title:      ZF/pair.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Ordered Pairs›

theory pair imports upair
begin

ML_file ‹simpdata.ML›

setup ‹
map_theory_simpset
(Simplifier.set_mksimps (fn ctxt => map mk_eq o ZF_atomize o Variable.gen_all ctxt)
›

ML ‹val ZF_ss = simpset_of \<^context>›

simproc_setup defined_Bex ("∃x∈A. P(x) ∧ Q(x)") = ‹
K (Quantifier1.rearrange_Bex (fn ctxt => unfold_tac ctxt @{thms Bex_def}))
›

simproc_setup defined_Ball ("∀x∈A. P(x) ⟶ Q(x)") = ‹
K (Quantifier1.rearrange_Ball (fn ctxt => unfold_tac ctxt @{thms Ball_def}))
›

(** Lemmas for showing that ⟨a,b⟩ uniquely determines a and b **)

lemma singleton_eq_iff [iff]: "{a} = {b} ⟷ a=b"
by (rule extension [THEN iff_trans], blast)

lemma doubleton_eq_iff: "{a,b} = {c,d} ⟷ (a=c ∧ b=d) | (a=d ∧ b=c)"
by (rule extension [THEN iff_trans], blast)

lemma Pair_iff [simp]: "⟨a,b⟩ = ⟨c,d⟩ ⟷ a=c ∧ b=d"
by (simp add: Pair_def doubleton_eq_iff, blast)

lemmas Pair_inject = Pair_iff [THEN iffD1, THEN conjE, elim!]

lemmas Pair_inject1 = Pair_iff [THEN iffD1, THEN conjunct1]
lemmas Pair_inject2 = Pair_iff [THEN iffD1, THEN conjunct2]

lemma Pair_not_0: "⟨a,b⟩ ≠ 0"
unfolding Pair_def
apply (blast elim: equalityE)
done

lemmas Pair_neq_0 = Pair_not_0 [THEN notE, elim!]

declare sym [THEN Pair_neq_0, elim!]

lemma Pair_neq_fst: "⟨a,b⟩=a ⟹ P"
proof (unfold Pair_def)
assume eq: "{{a, a}, {a, b}} = a"
have  "{a, a} ∈ {{a, a}, {a, b}}" by (rule consI1)
hence "{a, a} ∈ a" by (simp add: eq)
moreover have "a ∈ {a, a}" by (rule consI1)
ultimately show "P" by (rule mem_asym)
qed

lemma Pair_neq_snd: "⟨a,b⟩=b ⟹ P"
proof (unfold Pair_def)
assume eq: "{{a, a}, {a, b}} = b"
have  "{a, b} ∈ {{a, a}, {a, b}}" by blast
hence "{a, b} ∈ b" by (simp add: eq)
moreover have "b ∈ {a, b}" by blast
ultimately show "P" by (rule mem_asym)
qed

subsection‹Sigma: Disjoint Union of a Family of Sets›

text‹Generalizes Cartesian product›

lemma Sigma_iff [simp]: "⟨a,b⟩: Sigma(A,B) ⟷ a ∈ A ∧ b ∈ B(a)"

lemma SigmaI [TC,intro!]: "⟦a ∈ A;  b ∈ B(a)⟧ ⟹ ⟨a,b⟩ ∈ Sigma(A,B)"
by simp

lemmas SigmaD1 = Sigma_iff [THEN iffD1, THEN conjunct1]
lemmas SigmaD2 = Sigma_iff [THEN iffD1, THEN conjunct2]

(*The general elimination rule*)
lemma SigmaE [elim!]:
"⟦c ∈ Sigma(A,B);
⋀x y.⟦x ∈ A;  y ∈ B(x);  c=⟨x,y⟩⟧ ⟹ P
⟧ ⟹ P"
by (unfold Sigma_def, blast)

lemma SigmaE2 [elim!]:
"⟦⟨a,b⟩ ∈ Sigma(A,B);
⟦a ∈ A;  b ∈ B(a)⟧ ⟹ P
⟧ ⟹ P"
by (unfold Sigma_def, blast)

lemma Sigma_cong:
"⟦A=A';  ⋀x. x ∈ A' ⟹ B(x)=B'(x)⟧ ⟹
Sigma(A,B) = Sigma(A',B')"

(*Sigma_cong, Pi_cong NOT given to Addcongs: they cause
flex-flex pairs and the "Check your prover" error.  Most
Sigmas and Pis are abbreviated as * or -> *)

lemma Sigma_empty1 [simp]: "Sigma(0,B) = 0"
by blast

lemma Sigma_empty2 [simp]: "A*0 = 0"
by blast

lemma Sigma_empty_iff: "A*B=0 ⟷ A=0 | B=0"
by blast

subsection‹Projections \<^term>‹fst› and \<^term>‹snd››

lemma fst_conv [simp]: "fst(⟨a,b⟩) = a"

lemma snd_conv [simp]: "snd(⟨a,b⟩) = b"

lemma fst_type [TC]: "p ∈ Sigma(A,B) ⟹ fst(p) ∈ A"
by auto

lemma snd_type [TC]: "p ∈ Sigma(A,B) ⟹ snd(p) ∈ B(fst(p))"
by auto

lemma Pair_fst_snd_eq: "a ∈ Sigma(A,B) ⟹ <fst(a),snd(a)> = a"
by auto

subsection‹The Eliminator, \<^term>‹split››

(*A META-equality, so that it applies to higher types as well...*)
lemma split [simp]: "split(λx y. c(x,y), ⟨a,b⟩) ≡ c(a,b)"

lemma split_type [TC]:
"⟦p ∈ Sigma(A,B);
⋀x y.⟦x ∈ A; y ∈ B(x)⟧ ⟹ c(x,y):C(⟨x,y⟩)
⟧ ⟹ split(λx y. c(x,y), p) ∈ C(p)"
by (erule SigmaE, auto)

lemma expand_split:
"u ∈ A*B ⟹
R(split(c,u)) ⟷ (∀x∈A. ∀y∈B. u = ⟨x,y⟩ ⟶ R(c(x,y)))"

subsection‹A version of \<^term>‹split› for Formulae: Result Type \<^typ>‹o››

lemma splitI: "R(a,b) ⟹ split(R, ⟨a,b⟩)"

lemma splitE:
"⟦split(R,z);  z ∈ Sigma(A,B);
⋀x y. ⟦z = ⟨x,y⟩;  R(x,y)⟧ ⟹ P
⟧ ⟹ P"

lemma splitD: "split(R,⟨a,b⟩) ⟹ R(a,b)"

text ‹
\bigskip Complex rules for Sigma.
›

lemma split_paired_Bex_Sigma [simp]:
"(∃z ∈ Sigma(A,B). P(z)) ⟷ (∃x ∈ A. ∃y ∈ B(x). P(⟨x,y⟩))"
by blast

lemma split_paired_Ball_Sigma [simp]:
"(∀z ∈ Sigma(A,B). P(z)) ⟷ (∀x ∈ A. ∀y ∈ B(x). P(⟨x,y⟩))"
by blast

end

```