(* Title: ZF/sum.thy ID: $Id: Sum.thy,v 1.17 2005/06/17 14:15:09 haftmann 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 ML {* val sum_def = thm "sum_def"; val Inl_def = thm "Inl_def"; val Inr_def = thm "Inr_def"; val sum_defs = thms "sum_defs"; val Part_iff = thm "Part_iff"; val Part_eqI = thm "Part_eqI"; val PartI = thm "PartI"; val PartE = thm "PartE"; val Part_subset = thm "Part_subset"; val Sigma_bool = thm "Sigma_bool"; val InlI = thm "InlI"; val InrI = thm "InrI"; val sumE = thm "sumE"; val Inl_iff = thm "Inl_iff"; val Inr_iff = thm "Inr_iff"; val Inl_Inr_iff = thm "Inl_Inr_iff"; val Inr_Inl_iff = thm "Inr_Inl_iff"; val sum_empty = thm "sum_empty"; val Inl_inject = thm "Inl_inject"; val Inr_inject = thm "Inr_inject"; val Inl_neq_Inr = thm "Inl_neq_Inr"; val Inr_neq_Inl = thm "Inr_neq_Inl"; val InlD = thm "InlD"; val InrD = thm "InrD"; val sum_iff = thm "sum_iff"; val sum_subset_iff = thm "sum_subset_iff"; val sum_equal_iff = thm "sum_equal_iff"; val sum_eq_2_times = thm "sum_eq_2_times"; val case_Inl = thm "case_Inl"; val case_Inr = thm "case_Inr"; val case_type = thm "case_type"; val expand_case = thm "expand_case"; val case_cong = thm "case_cong"; val case_case = thm "case_case"; val Part_mono = thm "Part_mono"; val Part_Collect = thm "Part_Collect"; val Part_CollectE = thm "Part_CollectE"; val Part_Inl = thm "Part_Inl"; val Part_Inr = thm "Part_Inr"; val PartD1 = thm "PartD1"; val Part_id = thm "Part_id"; val Part_Inr2 = thm "Part_Inr2"; val Part_sum_equality = thm "Part_sum_equality"; *} 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)
lemmas PartI:
h(b) ∈ A ==> h(b) ∈ Part(A, h)
lemmas 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
lemmas sum_defs:
A + B == {0} × A ∪ {1} × B
Inl(a) == 〈0, a〉
Inr(b) == 〈1, b〉
case(c, d) == %〈y,z〉. cond(y, d(z), c(z))
lemmas sum_defs:
A + B == {0} × A ∪ {1} × B
Inl(a) == 〈0, a〉
Inr(b) == 〈1, b〉
case(c, d) == %〈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
lemmas Inl_inject:
Inl(a) = Inl(b) ==> a = b
lemmas Inl_inject:
Inl(a) = Inl(b) ==> a = b
lemmas Inr_inject:
Inr(a) = Inr(b) ==> a = b
lemmas Inr_inject:
Inr(a) = Inr(b) ==> a = b
lemmas Inl_neq_Inr:
Inl(a2) = Inr(b2) ==> P
lemmas Inl_neq_Inr:
Inl(a2) = Inr(b2) ==> P
lemmas Inr_neq_Inl:
Inr(b2) = Inl(a2) ==> P
lemmas 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(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) = case(%x. c(c'(x)), %y. 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)
lemmas Part_CollectE:
[| a ∈ Part({x ∈ A . P(x)}, h); [| a ∈ Part(A, h); P(a) |] ==> R |] ==> R
lemmas 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, %x. x) = A
lemma Part_Inr2:
Part(A + B, %x. Inr(h(x))) = {Inr(y) . y ∈ Part(B, h)}
lemma Part_sum_equality:
C ⊆ A + B ==> Part(C, Inl) ∪ Part(C, Inr) = C