(* Title: ZF/sum.thy ID: $Id: Sum.thy,v 1.18 2007/10/07 19:19:32 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) header{*Disjoint Sums*} theory Sum imports Bool equalities begin text{*And the "Part" primitive for simultaneous recursive type definitions*} global constdefs sum :: "[i,i]=>i" (infixr "+" 65) "A+B == {0}*A Un {1}*B" Inl :: "i=>i" "Inl(a) == <0,a>" Inr :: "i=>i" "Inr(b) == <1,b>" "case" :: "[i=>i, i=>i, i]=>i" "case(c,d) == (%<y,z>. cond(y, d(z), c(z)))" (*operator for selecting out the various summands*) Part :: "[i,i=>i] => i" "Part(A,h) == {x: A. EX z. x = h(z)}" local subsection{*Rules for the @{term Part} Primitive*} lemma Part_iff: "a : Part(A,h) <-> a:A & (EX y. a=h(y))" apply (unfold Part_def) apply (rule separation) done lemma Part_eqI [intro]: "[| a : A; a=h(b) |] ==> a : Part(A,h)" by (unfold Part_def, blast) lemmas PartI = refl [THEN [2] Part_eqI] lemma PartE [elim!]: "[| a : Part(A,h); !!z. [| a : A; a=h(z) |] ==> P |] ==> P" apply (unfold Part_def, blast) done lemma Part_subset: "Part(A,h) <= A" apply (unfold Part_def) apply (rule Collect_subset) done subsection{*Rules for Disjoint Sums*} lemmas sum_defs = sum_def Inl_def Inr_def case_def lemma Sigma_bool: "Sigma(bool,C) = C(0) + C(1)" by (unfold bool_def sum_def, blast) (** Introduction rules for the injections **) lemma InlI [intro!,simp,TC]: "a : A ==> Inl(a) : A+B" by (unfold sum_defs, blast) lemma InrI [intro!,simp,TC]: "b : B ==> Inr(b) : A+B" by (unfold sum_defs, blast) (** Elimination rules **) lemma sumE [elim!]: "[| u: A+B; !!x. [| x:A; u=Inl(x) |] ==> P; !!y. [| y:B; u=Inr(y) |] ==> P |] ==> P" by (unfold sum_defs, blast) (** Injection and freeness equivalences, for rewriting **) lemma Inl_iff [iff]: "Inl(a)=Inl(b) <-> a=b" by (simp add: sum_defs) lemma Inr_iff [iff]: "Inr(a)=Inr(b) <-> a=b" by (simp add: sum_defs) lemma Inl_Inr_iff [simp]: "Inl(a)=Inr(b) <-> False" by (simp add: sum_defs) lemma Inr_Inl_iff [simp]: "Inr(b)=Inl(a) <-> False" by (simp add: sum_defs) lemma sum_empty [simp]: "0+0 = 0" by (simp add: sum_defs) (*Injection and freeness rules*) lemmas Inl_inject = Inl_iff [THEN iffD1, standard] lemmas Inr_inject = Inr_iff [THEN iffD1, standard] lemmas Inl_neq_Inr = Inl_Inr_iff [THEN iffD1, THEN FalseE, elim!] lemmas Inr_neq_Inl = Inr_Inl_iff [THEN iffD1, THEN FalseE, elim!] lemma InlD: "Inl(a): A+B ==> a: A" by blast lemma InrD: "Inr(b): A+B ==> b: B" by blast lemma sum_iff: "u: A+B <-> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))" by blast lemma Inl_in_sum_iff [simp]: "(Inl(x) ∈ A+B) <-> (x ∈ A)"; by auto lemma Inr_in_sum_iff [simp]: "(Inr(y) ∈ A+B) <-> (y ∈ B)"; by auto lemma sum_subset_iff: "A+B <= C+D <-> A<=C & B<=D" by blast lemma sum_equal_iff: "A+B = C+D <-> A=C & B=D" by (simp add: extension sum_subset_iff, blast) lemma sum_eq_2_times: "A+A = 2*A" by (simp add: sum_def, blast) subsection{*The Eliminator: @{term case}*} lemma case_Inl [simp]: "case(c, d, Inl(a)) = c(a)" by (simp add: sum_defs) lemma case_Inr [simp]: "case(c, d, Inr(b)) = d(b)" by (simp add: sum_defs) lemma case_type [TC]: "[| u: A+B; !!x. x: A ==> c(x): C(Inl(x)); !!y. y: B ==> d(y): C(Inr(y)) |] ==> case(c,d,u) : C(u)" by auto lemma expand_case: "u: A+B ==> R(case(c,d,u)) <-> ((ALL x:A. u = Inl(x) --> R(c(x))) & (ALL y:B. u = Inr(y) --> R(d(y))))" by auto lemma case_cong: "[| z: A+B; !!x. x:A ==> c(x)=c'(x); !!y. y:B ==> d(y)=d'(y) |] ==> case(c,d,z) = case(c',d',z)" by auto lemma case_case: "z: A+B ==> case(c, d, case(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) = case(%x. c(c'(x)), %y. d(d'(y)), z)" by auto subsection{*More Rules for @{term "Part(A,h)"}*} lemma Part_mono: "A<=B ==> Part(A,h)<=Part(B,h)" by blast lemma Part_Collect: "Part(Collect(A,P), h) = Collect(Part(A,h), P)" by blast lemmas Part_CollectE = Part_Collect [THEN equalityD1, THEN subsetD, THEN CollectE, standard] lemma Part_Inl: "Part(A+B,Inl) = {Inl(x). x: A}" by blast lemma Part_Inr: "Part(A+B,Inr) = {Inr(y). y: B}" by blast lemma PartD1: "a : Part(A,h) ==> a : A" by (simp add: Part_def) lemma Part_id: "Part(A,%x. x) = A" by blast lemma Part_Inr2: "Part(A+B, %x. Inr(h(x))) = {Inr(y). y: Part(B,h)}" by blast lemma Part_sum_equality: "C <= A+B ==> Part(C,Inl) Un Part(C,Inr) = C" by blast end
lemma Part_iff:
a ∈ Part(A, h) <-> a ∈ A ∧ (∃y. a = h(y))
lemma Part_eqI:
[| a ∈ A; a = h(b) |] ==> a ∈ Part(A, h)
lemma PartI:
h(b) ∈ A ==> h(b) ∈ Part(A, h)
lemma PartE:
[| a ∈ Part(A, h); !!z. [| a ∈ A; a = h(z) |] ==> P |] ==> P
lemma Part_subset:
Part(A, h) ⊆ A
lemma sum_defs:
A + B == {0} × A ∪ {1} × B
Inl(a) == 〈0, a〉
Inr(b) == 〈1, b〉
case(c, d) == λ〈y,z〉. cond(y, d(z), c(z))
lemma Sigma_bool:
Sigma(bool, C) = C(0) + C(1)
lemma InlI:
a ∈ A ==> Inl(a) ∈ A + B
lemma InrI:
b ∈ B ==> Inr(b) ∈ A + B
lemma sumE:
[| u ∈ A + B; !!x. [| x ∈ A; u = Inl(x) |] ==> P;
!!y. [| y ∈ B; u = Inr(y) |] ==> P |]
==> P
lemma Inl_iff:
Inl(a) = Inl(b) <-> a = b
lemma Inr_iff:
Inr(a) = Inr(b) <-> a = b
lemma Inl_Inr_iff:
Inl(a) = Inr(b) <-> False
lemma Inr_Inl_iff:
Inr(b) = Inl(a) <-> False
lemma sum_empty:
0 + 0 = 0
lemma Inl_inject:
Inl(a) = Inl(b) ==> a = b
lemma Inr_inject:
Inr(a) = Inr(b) ==> a = b
lemma Inl_neq_Inr:
Inl(a2) = Inr(b2) ==> P
lemma Inr_neq_Inl:
Inr(b2) = Inl(a2) ==> P
lemma InlD:
Inl(a) ∈ A + B ==> a ∈ A
lemma InrD:
Inr(b) ∈ A + B ==> b ∈ B
lemma sum_iff:
u ∈ A + B <-> (∃x. x ∈ A ∧ u = Inl(x)) ∨ (∃y. y ∈ B ∧ u = Inr(y))
lemma Inl_in_sum_iff:
Inl(x) ∈ A + B <-> x ∈ A
lemma Inr_in_sum_iff:
Inr(y) ∈ A + B <-> y ∈ B
lemma sum_subset_iff:
A + B ⊆ C + D <-> A ⊆ C ∧ B ⊆ D
lemma sum_equal_iff:
A + B = C + D <-> A = C ∧ B = D
lemma sum_eq_2_times:
A + A = 2 × A
lemma case_Inl:
case(c, d, Inl(a)) = c(a)
lemma case_Inr:
case(c, d, Inr(b)) = d(b)
lemma case_type:
[| u ∈ A + B; !!x. x ∈ A ==> c(x) ∈ C(Inl(x));
!!y. y ∈ B ==> d(y) ∈ C(Inr(y)) |]
==> case(c, d, u) ∈ C(u)
lemma expand_case:
u ∈ A + B
==> R(case(c, d, u)) <->
(∀x∈A. u = Inl(x) --> R(c(x))) ∧ (∀y∈B. u = Inr(y) --> R(d(y)))
lemma case_cong:
[| z ∈ A + B; !!x. x ∈ A ==> c(x) = c'(x); !!y. y ∈ B ==> d(y) = d'(y) |]
==> case(c, d, z) = case(c', d', z)
lemma case_case:
z ∈ A + B
==> case(c, d, case(λx. Inl(c'(x)), λy. Inr(d'(y)), z)) =
case(λx. c(c'(x)), λy. d(d'(y)), z)
lemma Part_mono:
A ⊆ B ==> Part(A, h) ⊆ Part(B, h)
lemma Part_Collect:
Part(Collect(A, P), h) = Collect(Part(A, h), P)
lemma Part_CollectE:
[| a ∈ Part({x ∈ A . P(x)}, h); [| a ∈ Part(A, h); P(a) |] ==> R |] ==> R
lemma Part_Inl:
Part(A + B, Inl) = {Inl(x) . x ∈ A}
lemma Part_Inr:
Part(A + B, Inr) = {Inr(y) . y ∈ B}
lemma PartD1:
a ∈ Part(A, h) ==> a ∈ A
lemma Part_id:
Part(A, λx. x) = A
lemma Part_Inr2:
Part(A + B, λx. Inr(h(x))) = {Inr(y) . y ∈ Part(B, h)}
lemma Part_sum_equality:
C ⊆ A + B ==> Part(C, Inl) ∪ Part(C, Inr) = C