src/ZF/Sum.thy
 changeset 13240 bb5f4faea1f3 parent 13220 62c899c77151 child 13255 407ad9c3036d
1.1 --- a/src/ZF/Sum.thy	Sat Jun 22 18:28:46 2002 +0200
1.2 +++ b/src/ZF/Sum.thy	Sun Jun 23 10:14:13 2002 +0200
1.3 @@ -7,25 +7,270 @@
1.4  "Part" primitive for simultaneous recursive type definitions
1.5  *)
1.7 -Sum = Bool + equalities +
1.8 +theory Sum = Bool + equalities:
1.10  global
1.12 -consts
1.13 -    "+"     :: "[i,i]=>i"                     (infixr 65)
1.14 -    Inl     :: "i=>i"
1.15 -    Inr     :: "i=>i"
1.16 -    "case"  :: "[i=>i, i=>i, i]=>i"
1.17 -    Part    :: "[i,i=>i] => i"
1.18 +constdefs
1.19 +  sum     :: "[i,i]=>i"                     (infixr "+" 65)
1.20 +     "A+B == {0}*A Un {1}*B"
1.21 +
1.22 +  Inl     :: "i=>i"
1.23 +     "Inl(a) == <0,a>"
1.24 +
1.25 +  Inr     :: "i=>i"
1.26 +     "Inr(b) == <1,b>"
1.27 +
1.28 +  "case"  :: "[i=>i, i=>i, i]=>i"
1.29 +     "case(c,d) == (%<y,z>. cond(y, d(z), c(z)))"
1.30 +
1.31 +  (*operator for selecting out the various summands*)
1.32 +  Part    :: "[i,i=>i] => i"
1.33 +     "Part(A,h) == {x: A. EX z. x = h(z)}"
1.35  local
1.37 -defs
1.38 -    sum_def     "A+B == {0}*A Un {1}*B"
1.39 -    Inl_def     "Inl(a) == <0,a>"
1.40 -    Inr_def     "Inr(b) == <1,b>"
1.41 -    case_def    "case(c,d) == (%<y,z>. cond(y, d(z), c(z)))"
1.42 +(*** Rules for the Part primitive ***)
1.43 +
1.44 +lemma Part_iff:
1.45 +    "a : Part(A,h) <-> a:A & (EX y. a=h(y))"
1.46 +apply (unfold Part_def)
1.47 +apply (rule separation)
1.48 +done
1.49 +
1.50 +lemma Part_eqI [intro]:
1.51 +    "[| a : A;  a=h(b) |] ==> a : Part(A,h)"
1.52 +apply (unfold Part_def)
1.53 +apply blast
1.54 +done
1.55 +
1.56 +lemmas PartI = refl [THEN [2] Part_eqI]
1.57 +
1.58 +lemma PartE [elim!]:
1.59 +    "[| a : Part(A,h);  !!z. [| a : A;  a=h(z) |] ==> P
1.60 +     |] ==> P"
1.61 +apply (unfold Part_def)
1.62 +apply blast
1.63 +done
1.64 +
1.65 +lemma Part_subset: "Part(A,h) <= A"
1.66 +apply (unfold Part_def)
1.67 +apply (rule Collect_subset)
1.68 +done
1.69 +
1.70 +
1.71 +(*** Rules for Disjoint Sums ***)
1.72 +
1.73 +lemmas sum_defs = sum_def Inl_def Inr_def case_def
1.74 +
1.75 +lemma Sigma_bool: "Sigma(bool,C) = C(0) + C(1)"
1.76 +apply (unfold bool_def sum_def)
1.77 +apply blast
1.78 +done
1.79 +
1.80 +(** Introduction rules for the injections **)
1.81 +
1.82 +lemma InlI [intro!,simp,TC]: "a : A ==> Inl(a) : A+B"
1.83 +apply (unfold sum_defs)
1.84 +apply blast
1.85 +done
1.86 +
1.87 +lemma InrI [intro!,simp,TC]: "b : B ==> Inr(b) : A+B"
1.88 +apply (unfold sum_defs)
1.89 +apply blast
1.90 +done
1.91 +
1.92 +(** Elimination rules **)
1.93 +
1.94 +lemma sumE [elim!]:
1.95 +    "[| u: A+B;
1.96 +        !!x. [| x:A;  u=Inl(x) |] ==> P;
1.97 +        !!y. [| y:B;  u=Inr(y) |] ==> P
1.98 +     |] ==> P"
1.99 +apply (unfold sum_defs)
1.100 +apply (blast intro: elim:);
1.101 +done
1.103 +(** Injection and freeness equivalences, for rewriting **)
1.105 +lemma Inl_iff [iff]: "Inl(a)=Inl(b) <-> a=b"
1.107 +done
1.109 +lemma Inr_iff [iff]: "Inr(a)=Inr(b) <-> a=b"
1.111 +done
1.113 +lemma Inl_Inr_iff [iff]: "Inl(a)=Inr(b) <-> False"
1.115 +done
1.117 +lemma Inr_Inl_iff [iff]: "Inr(b)=Inl(a) <-> False"
1.119 +done
1.121 +lemma sum_empty [simp]: "0+0 = 0"
1.123 +done
1.125 +(*Injection and freeness rules*)
1.127 +lemmas Inl_inject = Inl_iff [THEN iffD1, standard]
1.128 +lemmas Inr_inject = Inr_iff [THEN iffD1, standard]
1.129 +lemmas Inl_neq_Inr = Inl_Inr_iff [THEN iffD1, THEN FalseE]
1.130 +lemmas Inr_neq_Inl = Inr_Inl_iff [THEN iffD1, THEN FalseE]
1.133 +lemma InlD: "Inl(a): A+B ==> a: A"
1.134 +apply blast
1.135 +done
1.137 +lemma InrD: "Inr(b): A+B ==> b: B"
1.138 +apply blast
1.139 +done
1.141 +lemma sum_iff: "u: A+B <-> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))"
1.142 +apply blast
1.143 +done
1.145 +lemma sum_subset_iff: "A+B <= C+D <-> A<=C & B<=D"
1.146 +apply blast
1.147 +done
1.149 +lemma sum_equal_iff: "A+B = C+D <-> A=C & B=D"
1.150 +apply (simp add: extension sum_subset_iff)
1.151 +apply blast
1.152 +done
1.154 +lemma sum_eq_2_times: "A+A = 2*A"
1.156 +apply blast
1.157 +done
1.160 +(*** Eliminator -- case ***)
1.162 -  (*operator for selecting out the various summands*)
1.163 -    Part_def    "Part(A,h) == {x: A. EX z. x = h(z)}"
1.164 +lemma case_Inl [simp]: "case(c, d, Inl(a)) = c(a)"
1.166 +done
1.168 +lemma case_Inr [simp]: "case(c, d, Inr(b)) = d(b)"
1.170 +done
1.172 +lemma case_type [TC]:
1.173 +    "[| u: A+B;
1.174 +        !!x. x: A ==> c(x): C(Inl(x));
1.175 +        !!y. y: B ==> d(y): C(Inr(y))
1.176 +     |] ==> case(c,d,u) : C(u)"
1.177 +apply (auto );
1.178 +done
1.180 +lemma expand_case: "u: A+B ==>
1.181 +        R(case(c,d,u)) <->
1.182 +        ((ALL x:A. u = Inl(x) --> R(c(x))) &
1.183 +        (ALL y:B. u = Inr(y) --> R(d(y))))"
1.184 +by auto
1.186 +lemma case_cong:
1.187 +  "[| z: A+B;
1.188 +      !!x. x:A ==> c(x)=c'(x);
1.189 +      !!y. y:B ==> d(y)=d'(y)
1.190 +   |] ==> case(c,d,z) = case(c',d',z)"
1.191 +by (auto );
1.193 +lemma case_case: "z: A+B ==>
1.194 +        case(c, d, case(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) =
1.195 +        case(%x. c(c'(x)), %y. d(d'(y)), z)"
1.196 +by auto
1.199 +(*** More rules for Part(A,h) ***)
1.201 +lemma Part_mono: "A<=B ==> Part(A,h)<=Part(B,h)"
1.202 +apply blast
1.203 +done
1.205 +lemma Part_Collect: "Part(Collect(A,P), h) = Collect(Part(A,h), P)"
1.206 +apply blast
1.207 +done
1.209 +lemmas Part_CollectE =
1.210 +     Part_Collect [THEN equalityD1, THEN subsetD, THEN CollectE, standard]
1.212 +lemma Part_Inl: "Part(A+B,Inl) = {Inl(x). x: A}"
1.213 +apply blast
1.214 +done
1.216 +lemma Part_Inr: "Part(A+B,Inr) = {Inr(y). y: B}"
1.217 +apply blast
1.218 +done
1.220 +lemma PartD1: "a : Part(A,h) ==> a : A"
1.222 +done
1.224 +lemma Part_id: "Part(A,%x. x) = A"
1.225 +apply blast
1.226 +done
1.228 +lemma Part_Inr2: "Part(A+B, %x. Inr(h(x))) = {Inr(y). y: Part(B,h)}"
1.229 +apply blast
1.230 +done
1.232 +lemma Part_sum_equality: "C <= A+B ==> Part(C,Inl) Un Part(C,Inr) = C"
1.233 +apply blast
1.234 +done
1.236 +ML
1.237 +{*
1.238 +val sum_def = thm "sum_def";
1.239 +val Inl_def = thm "Inl_def";
1.240 +val Inr_def = thm "Inr_def";
1.241 +val sum_defs = thms "sum_defs";
1.243 +val Part_iff = thm "Part_iff";
1.244 +val Part_eqI = thm "Part_eqI";
1.245 +val PartI = thm "PartI";
1.246 +val PartE = thm "PartE";
1.247 +val Part_subset = thm "Part_subset";
1.248 +val Sigma_bool = thm "Sigma_bool";
1.249 +val InlI = thm "InlI";
1.250 +val InrI = thm "InrI";
1.251 +val sumE = thm "sumE";
1.252 +val Inl_iff = thm "Inl_iff";
1.253 +val Inr_iff = thm "Inr_iff";
1.254 +val Inl_Inr_iff = thm "Inl_Inr_iff";
1.255 +val Inr_Inl_iff = thm "Inr_Inl_iff";
1.256 +val sum_empty = thm "sum_empty";
1.257 +val Inl_inject = thm "Inl_inject";
1.258 +val Inr_inject = thm "Inr_inject";
1.259 +val Inl_neq_Inr = thm "Inl_neq_Inr";
1.260 +val Inr_neq_Inl = thm "Inr_neq_Inl";
1.261 +val InlD = thm "InlD";
1.262 +val InrD = thm "InrD";
1.263 +val sum_iff = thm "sum_iff";
1.264 +val sum_subset_iff = thm "sum_subset_iff";
1.265 +val sum_equal_iff = thm "sum_equal_iff";
1.266 +val sum_eq_2_times = thm "sum_eq_2_times";
1.267 +val case_Inl = thm "case_Inl";
1.268 +val case_Inr = thm "case_Inr";
1.269 +val case_type = thm "case_type";
1.270 +val expand_case = thm "expand_case";
1.271 +val case_cong = thm "case_cong";
1.272 +val case_case = thm "case_case";
1.273 +val Part_mono = thm "Part_mono";
1.274 +val Part_Collect = thm "Part_Collect";
1.275 +val Part_CollectE = thm "Part_CollectE";
1.276 +val Part_Inl = thm "Part_Inl";
1.277 +val Part_Inr = thm "Part_Inr";
1.278 +val PartD1 = thm "PartD1";
1.279 +val Part_id = thm "Part_id";
1.280 +val Part_Inr2 = thm "Part_Inr2";
1.281 +val Part_sum_equality = thm "Part_sum_equality";
1.283 +*}
1.287  end