--- a/TODO Thu Dec 09 15:49:40 2004 +0100
+++ b/TODO Thu Dec 09 16:45:46 2004 +0100
@@ -14,11 +14,9 @@
who is interested)
- eliminate the last remaining old-style theories (Larry):
- Datatype_Universe.ML
HOL_lemmas.ML
NatArith.ML
Relation_Power.ML
- Sum_Type.ML
- update or remove ex/MT (Larry? Tobias?)
--- a/src/HOL/IsaMakefile Thu Dec 09 15:49:40 2004 +0100
+++ b/src/HOL/IsaMakefile Thu Dec 09 16:45:46 2004 +0100
@@ -98,7 +98,7 @@
Record.thy Relation.ML Relation.thy Relation_Power.ML \
Relation_Power.thy LOrder.thy OrderedGroup.thy OrderedGroup.ML Ring_and_Field.thy\
Set.ML Set.thy SetInterval.ML SetInterval.thy \
- Sum_Type.ML Sum_Type.thy Tools/datatype_abs_proofs.ML Tools/datatype_aux.ML \
+ Sum_Type.thy Tools/datatype_abs_proofs.ML Tools/datatype_aux.ML \
Tools/datatype_codegen.ML Tools/datatype_package.ML Tools/datatype_prop.ML \
Tools/datatype_realizer.ML Tools/datatype_rep_proofs.ML \
Tools/inductive_codegen.ML Tools/inductive_package.ML Tools/inductive_realizer.ML \
--- a/src/HOL/Sum_Type.ML Thu Dec 09 15:49:40 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,178 +0,0 @@
-(* Title: HOL/Sum_Type.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1991 University of Cambridge
-
-The disjoint sum of two types
-*)
-
-(** Inl_Rep and Inr_Rep: Representations of the constructors **)
-
-(*This counts as a non-emptiness result for admitting 'a+'b as a type*)
-Goalw [Sum_def] "Inl_Rep(a) : Sum";
-by (EVERY1 [rtac CollectI, rtac disjI1, rtac exI, rtac refl]);
-qed "Inl_RepI";
-
-Goalw [Sum_def] "Inr_Rep(b) : Sum";
-by (EVERY1 [rtac CollectI, rtac disjI2, rtac exI, rtac refl]);
-qed "Inr_RepI";
-
-Goal "inj_on Abs_Sum Sum";
-by (rtac inj_on_inverseI 1);
-by (etac Abs_Sum_inverse 1);
-qed "inj_on_Abs_Sum";
-
-(** Distinctness of Inl and Inr **)
-
-Goalw [Inl_Rep_def, Inr_Rep_def] "Inl_Rep(a) ~= Inr_Rep(b)";
-by (EVERY1 [rtac notI,
- etac (fun_cong RS fun_cong RS fun_cong RS iffE),
- rtac (notE RS ccontr), etac (mp RS conjunct2),
- REPEAT o (ares_tac [refl,conjI]) ]);
-qed "Inl_Rep_not_Inr_Rep";
-
-Goalw [Inl_def,Inr_def] "Inl(a) ~= Inr(b)";
-by (rtac (inj_on_Abs_Sum RS inj_on_contraD) 1);
-by (rtac Inl_Rep_not_Inr_Rep 1);
-by (rtac Inl_RepI 1);
-by (rtac Inr_RepI 1);
-qed "Inl_not_Inr";
-
-bind_thm ("Inr_not_Inl", Inl_not_Inr RS not_sym);
-
-AddIffs [Inl_not_Inr, Inr_not_Inl];
-
-bind_thm ("Inl_neq_Inr", Inl_not_Inr RS notE);
-bind_thm ("Inr_neq_Inl", sym RS Inl_neq_Inr);
-
-
-(** Injectiveness of Inl and Inr **)
-
-Goalw [Inl_Rep_def] "Inl_Rep(a) = Inl_Rep(c) ==> a=c";
-by (etac (fun_cong RS fun_cong RS fun_cong RS iffE) 1);
-by (Blast_tac 1);
-qed "Inl_Rep_inject";
-
-Goalw [Inr_Rep_def] "Inr_Rep(b) = Inr_Rep(d) ==> b=d";
-by (etac (fun_cong RS fun_cong RS fun_cong RS iffE) 1);
-by (Blast_tac 1);
-qed "Inr_Rep_inject";
-
-Goalw [Inl_def] "inj(Inl)";
-by (rtac injI 1);
-by (etac (inj_on_Abs_Sum RS inj_onD RS Inl_Rep_inject) 1);
-by (rtac Inl_RepI 1);
-by (rtac Inl_RepI 1);
-qed "inj_Inl";
-bind_thm ("Inl_inject", inj_Inl RS injD);
-
-Goalw [Inr_def] "inj(Inr)";
-by (rtac injI 1);
-by (etac (inj_on_Abs_Sum RS inj_onD RS Inr_Rep_inject) 1);
-by (rtac Inr_RepI 1);
-by (rtac Inr_RepI 1);
-qed "inj_Inr";
-bind_thm ("Inr_inject", inj_Inr RS injD);
-
-Goal "(Inl(x)=Inl(y)) = (x=y)";
-by (blast_tac (claset() addSDs [Inl_inject]) 1);
-qed "Inl_eq";
-
-Goal "(Inr(x)=Inr(y)) = (x=y)";
-by (blast_tac (claset() addSDs [Inr_inject]) 1);
-qed "Inr_eq";
-
-AddIffs [Inl_eq, Inr_eq];
-
-(*** Rules for the disjoint sum of two SETS ***)
-
-(** Introduction rules for the injections **)
-
-Goalw [sum_def] "a : A ==> Inl(a) : A <+> B";
-by (Blast_tac 1);
-qed "InlI";
-
-Goalw [sum_def] "b : B ==> Inr(b) : A <+> B";
-by (Blast_tac 1);
-qed "InrI";
-
-(** Elimination rules **)
-
-val major::prems = Goalw [sum_def]
- "[| u: A <+> B; \
-\ !!x. [| x:A; u=Inl(x) |] ==> P; \
-\ !!y. [| y:B; u=Inr(y) |] ==> P \
-\ |] ==> P";
-by (rtac (major RS UnE) 1);
-by (REPEAT (rtac refl 1
- ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
-qed "PlusE";
-
-
-AddSIs [InlI, InrI];
-AddSEs [PlusE];
-
-
-(** Exhaustion rule for sums -- a degenerate form of induction **)
-
-val prems = Goalw [Inl_def,Inr_def]
- "[| !!x::'a. s = Inl(x) ==> P; !!y::'b. s = Inr(y) ==> P \
-\ |] ==> P";
-by (rtac (rewrite_rule [Sum_def] Rep_Sum RS CollectE) 1);
-by (REPEAT (eresolve_tac [disjE,exE] 1
- ORELSE EVERY1 [resolve_tac prems,
- etac subst,
- rtac (Rep_Sum_inverse RS sym)]));
-qed "sumE";
-
-val prems = Goal "[| !!x. P (Inl x); !!x. P (Inr x) |] ==> P x";
-by (res_inst_tac [("s","x")] sumE 1);
-by (ALLGOALS (hyp_subst_tac THEN' (resolve_tac prems)));
-qed "sum_induct";
-
-
-(** Rules for the Part primitive **)
-
-Goalw [Part_def] "[| a : A; a=h(b) |] ==> a : Part A h";
-by (Blast_tac 1);
-qed "Part_eqI";
-
-bind_thm ("PartI", refl RSN (2,Part_eqI));
-
-val major::prems = Goalw [Part_def]
- "[| a : Part A h; !!z. [| a : A; a=h(z) |] ==> P \
-\ |] ==> P";
-by (rtac (major RS IntE) 1);
-by (etac CollectE 1);
-by (etac exE 1);
-by (REPEAT (ares_tac prems 1));
-qed "PartE";
-
-AddIs [Part_eqI];
-AddSEs [PartE];
-
-Goalw [Part_def] "Part A h <= A";
-by (rtac Int_lower1 1);
-qed "Part_subset";
-
-Goal "A<=B ==> Part A h <= Part B h";
-by (Blast_tac 1);
-qed "Part_mono";
-
-bind_thms ("basic_monos", basic_monos @ [Part_mono]);
-
-Goalw [Part_def] "a : Part A h ==> a : A";
-by (etac IntD1 1);
-qed "PartD1";
-
-Goal "Part A (%x. x) = A";
-by (Blast_tac 1);
-qed "Part_id";
-
-Goal "Part (A Int B) h = (Part A h) Int (Part B h)";
-by (Blast_tac 1);
-qed "Part_Int";
-
-Goal "Part (A Int {x. P x}) h = (Part A h) Int {x. P x}";
-by (Blast_tac 1);
-qed "Part_Collect";
--- a/src/HOL/Sum_Type.thy Thu Dec 09 15:49:40 2004 +0100
+++ b/src/HOL/Sum_Type.thy Thu Dec 09 16:45:46 2004 +0100
@@ -2,47 +2,221 @@
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
-
-The disjoint sum of two types.
*)
-Sum_Type = Product_Type +
+header{*The Disjoint Sum of Two Types*}
-(* type definition *)
+theory Sum_Type
+imports Product_Type
+begin
+
+text{*The representations of the two injections*}
constdefs
- Inl_Rep :: ['a, 'a, 'b, bool] => bool
+ Inl_Rep :: "['a, 'a, 'b, bool] => bool"
"Inl_Rep == (%a. %x y p. x=a & p)"
- Inr_Rep :: ['b, 'a, 'b, bool] => bool
+ Inr_Rep :: "['b, 'a, 'b, bool] => bool"
"Inr_Rep == (%b. %x y p. y=b & ~p)"
+
global
typedef (Sum)
('a, 'b) "+" (infixr 10)
= "{f. (? a. f = Inl_Rep(a::'a)) | (? b. f = Inr_Rep(b::'b))}"
-
-
-(* abstract constants and syntax *)
-
-consts
- Inl :: "'a => 'a + 'b"
- Inr :: "'b => 'a + 'b"
-
- (*disjoint sum for sets; the operator + is overloaded with wrong type!*)
- Plus :: "['a set, 'b set] => ('a + 'b) set" (infixr "<+>" 65)
- Part :: ['a set, 'b => 'a] => 'a set
+ by auto
local
-defs
- Inl_def "Inl == (%a. Abs_Sum(Inl_Rep(a)))"
- Inr_def "Inr == (%b. Abs_Sum(Inr_Rep(b)))"
+
+text{*abstract constants and syntax*}
+
+constdefs
+ Inl :: "'a => 'a + 'b"
+ "Inl == (%a. Abs_Sum(Inl_Rep(a)))"
+
+ Inr :: "'b => 'a + 'b"
+ "Inr == (%b. Abs_Sum(Inr_Rep(b)))"
+
+ Plus :: "['a set, 'b set] => ('a + 'b) set" (infixr "<+>" 65)
+ "A <+> B == (Inl`A) Un (Inr`B)"
+ --{*disjoint sum for sets; the operator + is overloaded with wrong type!*}
+
+ Part :: "['a set, 'b => 'a] => 'a set"
+ "Part A h == A Int {x. ? z. x = h(z)}"
+ --{*for selecting out the components of a mutually recursive definition*}
+
+
+
+(** Inl_Rep and Inr_Rep: Representations of the constructors **)
+
+(*This counts as a non-emptiness result for admitting 'a+'b as a type*)
+lemma Inl_RepI: "Inl_Rep(a) : Sum"
+by (auto simp add: Sum_def)
+
+lemma Inr_RepI: "Inr_Rep(b) : Sum"
+by (auto simp add: Sum_def)
+
+lemma inj_on_Abs_Sum: "inj_on Abs_Sum Sum"
+apply (rule inj_on_inverseI)
+apply (erule Abs_Sum_inverse)
+done
+
+subsection{*Freeness Properties for @{term Inl} and @{term Inr}*}
+
+text{*Distinctness*}
+
+lemma Inl_Rep_not_Inr_Rep: "Inl_Rep(a) ~= Inr_Rep(b)"
+by (auto simp add: Inl_Rep_def Inr_Rep_def expand_fun_eq)
+
+lemma Inl_not_Inr [iff]: "Inl(a) ~= Inr(b)"
+apply (simp add: Inl_def Inr_def)
+apply (rule inj_on_Abs_Sum [THEN inj_on_contraD])
+apply (rule Inl_Rep_not_Inr_Rep)
+apply (rule Inl_RepI)
+apply (rule Inr_RepI)
+done
+
+lemmas Inr_not_Inl = Inl_not_Inr [THEN not_sym, standard, iff]
+
+lemmas Inl_neq_Inr = Inl_not_Inr [THEN notE, standard]
+lemmas Inr_neq_Inl = sym [THEN Inl_neq_Inr, standard]
+
+
+text{*Injectiveness*}
+
+lemma Inl_Rep_inject: "Inl_Rep(a) = Inl_Rep(c) ==> a=c"
+by (auto simp add: Inl_Rep_def expand_fun_eq)
+
+lemma Inr_Rep_inject: "Inr_Rep(b) = Inr_Rep(d) ==> b=d"
+by (auto simp add: Inr_Rep_def expand_fun_eq)
+
+lemma inj_Inl: "inj(Inl)"
+apply (simp add: Inl_def)
+apply (rule inj_onI)
+apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inl_Rep_inject])
+apply (rule Inl_RepI)
+apply (rule Inl_RepI)
+done
+lemmas Inl_inject = inj_Inl [THEN injD, standard]
+
+lemma inj_Inr: "inj(Inr)"
+apply (simp add: Inr_def)
+apply (rule inj_onI)
+apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inr_Rep_inject])
+apply (rule Inr_RepI)
+apply (rule Inr_RepI)
+done
+
+lemmas Inr_inject = inj_Inr [THEN injD, standard]
+
+lemma Inl_eq [iff]: "(Inl(x)=Inl(y)) = (x=y)"
+by (blast dest!: Inl_inject)
+
+lemma Inr_eq [iff]: "(Inr(x)=Inr(y)) = (x=y)"
+by (blast dest!: Inr_inject)
+
+
+subsection{*The Disjoint Sum of Sets*}
+
+(** Introduction rules for the injections **)
- sum_def "A <+> B == (Inl`A) Un (Inr`B)"
+lemma InlI [intro!]: "a : A ==> Inl(a) : A <+> B"
+by (simp add: Plus_def)
+
+lemma InrI [intro!]: "b : B ==> Inr(b) : A <+> B"
+by (simp add: Plus_def)
+
+(** Elimination rules **)
+
+lemma PlusE [elim!]:
+ "[| u: A <+> B;
+ !!x. [| x:A; u=Inl(x) |] ==> P;
+ !!y. [| y:B; u=Inr(y) |] ==> P
+ |] ==> P"
+by (auto simp add: Plus_def)
+
+
+
+text{*Exhaustion rule for sums, a degenerate form of induction*}
+lemma sumE:
+ "[| !!x::'a. s = Inl(x) ==> P; !!y::'b. s = Inr(y) ==> P
+ |] ==> P"
+apply (rule Abs_Sum_cases [of s])
+apply (auto simp add: Sum_def Inl_def Inr_def)
+done
+
+lemma sum_induct: "[| !!x. P (Inl x); !!x. P (Inr x) |] ==> P x"
+by (rule sumE [of x], auto)
+
+
+subsection{*The @{term Part} Primitive*}
+
+lemma Part_eqI [intro]: "[| a : A; a=h(b) |] ==> a : Part A h"
+by (auto simp add: Part_def)
+
+lemmas PartI = Part_eqI [OF _ refl, standard]
+
+lemma PartE [elim!]: "[| a : Part A h; !!z. [| a : A; a=h(z) |] ==> P |] ==> P"
+by (auto simp add: Part_def)
+
+
+lemma Part_subset: "Part A h <= A"
+by (auto simp add: Part_def)
+
+lemma Part_mono: "A<=B ==> Part A h <= Part B h"
+by blast
+
+lemmas basic_monos = basic_monos Part_mono
- (*for selecting out the components of a mutually recursive definition*)
- Part_def "Part A h == A Int {x. ? z. x = h(z)}"
+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_Int: "Part (A Int B) h = (Part A h) Int (Part B h)"
+by blast
+
+lemma Part_Collect: "Part (A Int {x. P x}) h = (Part A h) Int {x. P x}"
+by blast
+
+ML
+{*
+val Inl_RepI = thm "Inl_RepI";
+val Inr_RepI = thm "Inr_RepI";
+val inj_on_Abs_Sum = thm "inj_on_Abs_Sum";
+val Inl_Rep_not_Inr_Rep = thm "Inl_Rep_not_Inr_Rep";
+val Inl_not_Inr = thm "Inl_not_Inr";
+val Inr_not_Inl = thm "Inr_not_Inl";
+val Inl_neq_Inr = thm "Inl_neq_Inr";
+val Inr_neq_Inl = thm "Inr_neq_Inl";
+val Inl_Rep_inject = thm "Inl_Rep_inject";
+val Inr_Rep_inject = thm "Inr_Rep_inject";
+val inj_Inl = thm "inj_Inl";
+val Inl_inject = thm "Inl_inject";
+val inj_Inr = thm "inj_Inr";
+val Inr_inject = thm "Inr_inject";
+val Inl_eq = thm "Inl_eq";
+val Inr_eq = thm "Inr_eq";
+val InlI = thm "InlI";
+val InrI = thm "InrI";
+val PlusE = thm "PlusE";
+val sumE = thm "sumE";
+val sum_induct = thm "sum_induct";
+val Part_eqI = thm "Part_eqI";
+val PartI = thm "PartI";
+val PartE = thm "PartE";
+val Part_subset = thm "Part_subset";
+val Part_mono = thm "Part_mono";
+val PartD1 = thm "PartD1";
+val Part_id = thm "Part_id";
+val Part_Int = thm "Part_Int";
+val Part_Collect = thm "Part_Collect";
+
+val basic_monos = thms "basic_monos";
+*}
+
end
--- a/src/HOL/Tools/datatype_rep_proofs.ML Thu Dec 09 15:49:40 2004 +0100
+++ b/src/HOL/Tools/datatype_rep_proofs.ML Thu Dec 09 16:45:46 2004 +0100
@@ -103,9 +103,9 @@
val Type (_, [T1, T2]) = T
in
if i <= n2 then
- Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
+ Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
else
- Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
+ Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
end
in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
end;
--- a/src/HOL/Tools/inductive_package.ML Thu Dec 09 15:49:40 2004 +0100
+++ b/src/HOL/Tools/inductive_package.ML Thu Dec 09 16:45:46 2004 +0100
@@ -212,9 +212,9 @@
val Type (_, [T1, T2]) = T
in
if i <= n2 then
- Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
+ Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
else
- Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
+ Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
end
in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
end;