(* Title: ZF/pair ID: $Id: pair.thy,v 1.10 2005/08/02 17:47:11 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header{*Ordered Pairs*} theory pair imports upair uses "simpdata.ML" begin (** 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, standard, elim!] lemmas Pair_inject1 = Pair_iff [THEN iffD1, THEN conjunct1, standard] lemmas Pair_inject2 = Pair_iff [THEN iffD1, THEN conjunct2, standard] lemma Pair_not_0: "<a,b> ~= 0" apply (unfold Pair_def) apply (blast elim: equalityE) done lemmas Pair_neq_0 = Pair_not_0 [THEN notE, standard, elim!] declare sym [THEN Pair_neq_0, elim!] lemma Pair_neq_fst: "<a,b>=a ==> P" apply (unfold Pair_def) apply (rule consI1 [THEN mem_asym, THEN FalseE]) apply (erule subst) apply (rule consI1) done lemma Pair_neq_snd: "<a,b>=b ==> P" apply (unfold Pair_def) apply (rule consI1 [THEN consI2, THEN mem_asym, THEN FalseE]) apply (erule subst) apply (rule consI1 [THEN consI2]) done 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)" by (simp add: Sigma_def) lemma SigmaI [TC,intro!]: "[| a:A; b:B(a) |] ==> <a,b> : Sigma(A,B)" by simp lemmas SigmaD1 = Sigma_iff [THEN iffD1, THEN conjunct1, standard] lemmas SigmaD2 = Sigma_iff [THEN iffD1, THEN conjunct2, standard] (*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')" by (simp add: Sigma_def) (*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" by (simp add: fst_def) lemma snd_conv [simp]: "snd(<a,b>) = b" by (simp add: snd_def) 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(%x y. c(x,y), <a,b>) == c(a,b)" by (simp add: split_def) lemma split_type [TC]: "[| p:Sigma(A,B); !!x y.[| x:A; y:B(x) |] ==> c(x,y):C(<x,y>) |] ==> split(%x y. c(x,y), p) : C(p)" apply (erule SigmaE, auto) done lemma expand_split: "u: A*B ==> R(split(c,u)) <-> (ALL x:A. ALL y:B. u = <x,y> --> R(c(x,y)))" apply (simp add: split_def) apply auto done subsection{*A version of @{term split} for Formulae: Result Type @{typ o}*} lemma splitI: "R(a,b) ==> split(R, <a,b>)" by (simp add: split_def) lemma splitE: "[| split(R,z); z:Sigma(A,B); !!x y. [| z = <x,y>; R(x,y) |] ==> P |] ==> P" apply (simp add: split_def) apply (erule SigmaE, force) done lemma splitD: "split(R,<a,b>) ==> R(a,b)" by (simp add: split_def) 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 ML {* val singleton_eq_iff = thm "singleton_eq_iff"; val doubleton_eq_iff = thm "doubleton_eq_iff"; val Pair_iff = thm "Pair_iff"; val Pair_inject = thm "Pair_inject"; val Pair_inject1 = thm "Pair_inject1"; val Pair_inject2 = thm "Pair_inject2"; val Pair_not_0 = thm "Pair_not_0"; val Pair_neq_0 = thm "Pair_neq_0"; val Pair_neq_fst = thm "Pair_neq_fst"; val Pair_neq_snd = thm "Pair_neq_snd"; val Sigma_iff = thm "Sigma_iff"; val SigmaI = thm "SigmaI"; val SigmaD1 = thm "SigmaD1"; val SigmaD2 = thm "SigmaD2"; val SigmaE = thm "SigmaE"; val SigmaE2 = thm "SigmaE2"; val Sigma_cong = thm "Sigma_cong"; val Sigma_empty1 = thm "Sigma_empty1"; val Sigma_empty2 = thm "Sigma_empty2"; val Sigma_empty_iff = thm "Sigma_empty_iff"; val fst_conv = thm "fst_conv"; val snd_conv = thm "snd_conv"; val fst_type = thm "fst_type"; val snd_type = thm "snd_type"; val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; val split = thm "split"; val split_type = thm "split_type"; val expand_split = thm "expand_split"; val splitI = thm "splitI"; val splitE = thm "splitE"; val splitD = thm "splitD"; *} end
lemma singleton_eq_iff:
{a} = {b} <-> a = b
lemma doubleton_eq_iff:
{a, b} = {c, d} <-> a = c ∧ b = d ∨ a = d ∧ b = c
lemma Pair_iff:
〈a, b〉 = 〈c, d〉 <-> a = c ∧ b = d
lemmas Pair_inject:
[| 〈a, b〉 = 〈c, d〉; [| a = c; b = d |] ==> R |] ==> R
lemmas Pair_inject:
[| 〈a, b〉 = 〈c, d〉; [| a = c; b = d |] ==> R |] ==> R
lemmas Pair_inject1:
〈a, b〉 = 〈c, d〉 ==> a = c
lemmas Pair_inject1:
〈a, b〉 = 〈c, d〉 ==> a = c
lemmas Pair_inject2:
〈a, b〉 = 〈c, d〉 ==> b = d
lemmas Pair_inject2:
〈a, b〉 = 〈c, d〉 ==> b = d
lemma Pair_not_0:
〈a, b〉 ≠ 0
lemmas Pair_neq_0:
〈a, b〉 = 0 ==> R
lemmas Pair_neq_0:
〈a, b〉 = 0 ==> R
lemma Pair_neq_fst:
〈a, b〉 = a ==> P
lemma Pair_neq_snd:
〈a, b〉 = b ==> P
lemma Sigma_iff:
〈a, b〉 ∈ Sigma(A, B) <-> a ∈ A ∧ b ∈ B(a)
lemma SigmaI:
[| a ∈ A; b ∈ B(a) |] ==> 〈a, b〉 ∈ Sigma(A, B)
lemmas SigmaD1:
〈a, b〉 ∈ Sigma(A, B) ==> a ∈ A
lemmas SigmaD1:
〈a, b〉 ∈ Sigma(A, B) ==> a ∈ A
lemmas SigmaD2:
〈a, b〉 ∈ Sigma(A, B) ==> b ∈ B(a)
lemmas SigmaD2:
〈a, b〉 ∈ Sigma(A, B) ==> b ∈ B(a)
lemma SigmaE:
[| c ∈ Sigma(A, B); !!x y. [| x ∈ A; y ∈ B(x); c = 〈x, y〉 |] ==> P |] ==> P
lemma SigmaE2:
[| 〈a, b〉 ∈ Sigma(A, B); [| a ∈ A; b ∈ B(a) |] ==> P |] ==> P
lemma Sigma_cong:
[| A = A'; !!x. x ∈ A' ==> B(x) = B'(x) |] ==> Sigma(A, B) = Sigma(A', B')
lemma Sigma_empty1:
Sigma(0, B) = 0
lemma Sigma_empty2:
A × 0 = 0
lemma Sigma_empty_iff:
A × B = 0 <-> A = 0 ∨ B = 0
lemma fst_conv:
fst(〈a, b〉) = a
lemma snd_conv:
snd(〈a, b〉) = b
lemma fst_type:
p ∈ Sigma(A, B) ==> fst(p) ∈ A
lemma snd_type:
p ∈ Sigma(A, B) ==> snd(p) ∈ B(fst(p))
lemma Pair_fst_snd_eq:
a ∈ Sigma(A, B) ==> 〈fst(a), snd(a)〉 = a
lemma split:
(%〈x,y〉. c(x, y))(〈a, b〉) == c(a, b)
lemma split_type:
[| p ∈ Sigma(A, B); !!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) ∈ C(〈x, y〉) |] ==> (%〈x,y〉. c(x, y))(p) ∈ C(p)
lemma expand_split:
u ∈ A × B ==> R(split(c, u)) <-> (∀x∈A. ∀y∈B. u = 〈x, y〉 --> R(c(x, y)))
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)
lemma split_paired_Bex_Sigma:
(∃z∈Sigma(A, B). P(z)) <-> (∃x∈A. ∃y∈B(x). P(〈x, y〉))
lemma split_paired_Ball_Sigma:
(∀z∈Sigma(A, B). P(z)) <-> (∀x∈A. ∀y∈B(x). P(〈x, y〉))