diff --git a/src/CTT/CTT.thy b/src/CTT/CTT.thy
--- a/src/CTT/CTT.thy
+++ b/src/CTT/CTT.thy
@@ -1,952 +1,952 @@
 (*  Title:      CTT/CTT.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1993  University of Cambridge
 *)
 
 theory CTT
 imports Pure
 begin
 
 section \<open>Constructive Type Theory: axiomatic basis\<close>
 
 ML_file \<open>~~/src/Provers/typedsimp.ML\<close>
 setup Pure_Thy.old_appl_syntax_setup
 
 typedecl i
 typedecl t
 typedecl o
 
 consts
   \<comment> \<open>Types\<close>
   F         :: "t"
   T         :: "t"          \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close>
   contr     :: "i\<Rightarrow>i"
   tt        :: "i"
   \<comment> \<open>Natural numbers\<close>
   N         :: "t"
   succ      :: "i\<Rightarrow>i"
   rec       :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i"
   \<comment> \<open>Unions\<close>
   inl       :: "i\<Rightarrow>i"
   inr       :: "i\<Rightarrow>i"
   "when"    :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i"
   \<comment> \<open>General Sum and Binary Product\<close>
   Sum       :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
   fst       :: "i\<Rightarrow>i"
   snd       :: "i\<Rightarrow>i"
   split     :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i"
   \<comment> \<open>General Product and Function Space\<close>
   Prod      :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
   \<comment> \<open>Types\<close>
   Plus      :: "[t,t]\<Rightarrow>t"           (infixr "+" 40)
   \<comment> \<open>Equality type\<close>
   Eq        :: "[t,i,i]\<Rightarrow>t"
   eq        :: "i"
   \<comment> \<open>Judgements\<close>
   Type      :: "t \<Rightarrow> prop"          ("(_ type)" [10] 5)
   Eqtype    :: "[t,t]\<Rightarrow>prop"        ("(_ =/ _)" [10,10] 5)
   Elem      :: "[i, t]\<Rightarrow>prop"       ("(_ /: _)" [10,10] 5)
   Eqelem    :: "[i,i,t]\<Rightarrow>prop"      ("(_ =/ _ :/ _)" [10,10,10] 5)
   Reduce    :: "[i,i]\<Rightarrow>prop"        ("Reduce[_,_]")
 
   \<comment> \<open>Types\<close>
 
   \<comment> \<open>Functions\<close>
   lambda    :: "(i \<Rightarrow> i) \<Rightarrow> i"      (binder "\<^bold>\<lambda>" 10)
   app       :: "[i,i]\<Rightarrow>i"           (infixl "`" 60)
   \<comment> \<open>Natural numbers\<close>
   Zero      :: "i"                  ("0")
   \<comment> \<open>Pairing\<close>
   pair      :: "[i,i]\<Rightarrow>i"           ("(1<_,/_>)")
 
 syntax
   "_PROD"   :: "[idt,t,t]\<Rightarrow>t"       ("(3\<Prod>_:_./ _)" 10)
   "_SUM"    :: "[idt,t,t]\<Rightarrow>t"       ("(3\<Sum>_:_./ _)" 10)
 translations
   "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)"
   "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)"
 
 abbreviation Arrow :: "[t,t]\<Rightarrow>t"  (infixr "\<longrightarrow>" 30)
   where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B"
 
 abbreviation Times :: "[t,t]\<Rightarrow>t"  (infixr "\<times>" 50)
   where "A \<times> B \<equiv> \<Sum>_:A. B"
 
 text \<open>
   Reduction: a weaker notion than equality;  a hack for simplification.
   \<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else
     that \<open>a\<close> and \<open>b\<close> are textually identical.
 
   Does not verify \<open>a:A\<close>!  Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close>
   premise. No new theorems can be proved about the standard judgements.
 \<close>
 axiomatization
 where
   refl_red: "\<And>a. Reduce[a,a]" and
   red_if_equal: "\<And>a b A. a = b : A \<Longrightarrow> Reduce[a,b]" and
   trans_red: "\<And>a b c A. \<lbrakk>a = b : A; Reduce[b,c]\<rbrakk> \<Longrightarrow> a = c : A" and
 
   \<comment> \<open>Reflexivity\<close>
 
   refl_type: "\<And>A. A type \<Longrightarrow> A = A" and
   refl_elem: "\<And>a A. a : A \<Longrightarrow> a = a : A" and
 
   \<comment> \<open>Symmetry\<close>
 
   sym_type:  "\<And>A B. A = B \<Longrightarrow> B = A" and
   sym_elem:  "\<And>a b A. a = b : A \<Longrightarrow> b = a : A" and
 
   \<comment> \<open>Transitivity\<close>
 
   trans_type:   "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and
   trans_elem:   "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and
 
   equal_types:  "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and
   equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and
 
   \<comment> \<open>Substitution\<close>
 
   subst_type:   "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and
   subst_typeL:  "\<And>a c A B D. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> B(z) = D(z)\<rbrakk> \<Longrightarrow> B(a) = D(c)" and
 
   subst_elem:   "\<And>a b A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> b(z):B(z)\<rbrakk> \<Longrightarrow> b(a):B(a)" and
   subst_elemL:
     "\<And>a b c d A B. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> b(z)=d(z) : B(z)\<rbrakk> \<Longrightarrow> b(a)=d(c) : B(a)" and
 
 
   \<comment> \<open>The type \<open>N\<close> -- natural numbers\<close>
 
   NF: "N type" and
   NI0: "0 : N" and
   NI_succ: "\<And>a. a : N \<Longrightarrow> succ(a) : N" and
   NI_succL:  "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and
 
   NE:
    "\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
    \<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and
 
   NEL:
    "\<And>p q a b c d C. \<lbrakk>p = q : N; a = c : C(0);
       \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk>
    \<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) = rec(q,c,d) : C(p)" and
 
   NC0:
    "\<And>a b C. \<lbrakk>a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
    \<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and
 
   NC_succ:
    "\<And>p a b C. \<lbrakk>p: N;  a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> \<Longrightarrow>
    rec(succ(p), a, \<lambda>u v. b(u,v)) = b(p, rec(p, a, \<lambda>u v. b(u,v))) : C(succ(p))" and
 
   \<comment> \<open>The fourth Peano axiom.  See page 91 of Martin-Löf's book.\<close>
   zero_ne_succ: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and
 
 
   \<comment> \<open>The Product of a family of types\<close>
 
   ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) type" and
 
   ProdFL:
     "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) = \<Prod>x:C. D(x)" and
 
   ProdI:
     "\<And>b A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x):B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and
 
   ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow>
     \<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and
 
   ProdE:  "\<And>p a A B. \<lbrakk>p : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and
   ProdEL: "\<And>p q a b A B. \<lbrakk>p = q: \<Prod>x:A. B(x); a = b : A\<rbrakk> \<Longrightarrow> p`a = q`b : B(a)" and
 
   ProdC: "\<And>a b A B. \<lbrakk>a : A; \<And>x. x:A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x)) ` a = b(a) : B(a)" and
 
   ProdC2: "\<And>p A B. p : \<Prod>x:A. B(x) \<Longrightarrow> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and
 
 
   \<comment> \<open>The Sum of a family of types\<close>
 
   SumF:  "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) type" and
   SumFL: "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) = \<Sum>x:C. D(x)" and
 
   SumI:  "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)" and
   SumIL: "\<And>a b c d A B. \<lbrakk> a = c : A; b = d : B(a)\<rbrakk> \<Longrightarrow> <a,b> = <c,d> : \<Sum>x:A. B(x)" and
 
   SumE: "\<And>p c A B C. \<lbrakk>p: \<Sum>x:A. B(x); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
     \<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and
 
   SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : \<Sum>x:A. B(x);
       \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk>
     \<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and
 
   SumC: "\<And>a b c A B C. \<lbrakk>a: A;  b: B(a); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
     \<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and
 
   fst_def:   "\<And>a. fst(a) \<equiv> split(a, \<lambda>x y. x)" and
   snd_def:   "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and
 
 
   \<comment> \<open>The sum of two types\<close>
 
   PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and
   PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and
 
   PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and
   PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and
 
   PlusI_inr: "\<And>b A B. \<lbrakk>A type; b : B\<rbrakk> \<Longrightarrow> inr(b) : A+B" and
   PlusI_inrL: "\<And>b d A B. \<lbrakk>A type; b = d : B\<rbrakk> \<Longrightarrow> inr(b) = inr(d) : A+B" and
 
   PlusE:
     "\<And>p c d A B C. \<lbrakk>p: A+B;
       \<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
       \<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) : C(p)" and
 
   PlusEL:
     "\<And>p q c d e f A B C. \<lbrakk>p = q : A+B;
       \<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x));
       \<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk>
     \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and
 
   PlusC_inl:
     "\<And>a c d A B C. \<lbrakk>a: A;
       \<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
       \<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk>
     \<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and
 
   PlusC_inr:
     "\<And>b c d A B C. \<lbrakk>b: B;
       \<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
       \<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk>
     \<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and
 
 
   \<comment> \<open>The type \<open>Eq\<close>\<close>
 
   EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and
   EqFL: "\<And>a b c d A B. \<lbrakk>A = B; a = c : A; b = d : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) = Eq(B,c,d)" and
   EqI: "\<And>a b A. a = b : A \<Longrightarrow> eq : Eq(A,a,b)" and
   EqE: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> a = b : A" and
 
   \<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close>
   EqC: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> p = eq : Eq(A,a,b)" and
 
 
   \<comment> \<open>The type \<open>F\<close>\<close>
 
   FF: "F type" and
   FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and
   FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> contr(p) = contr(q) : C" and
 
 
   \<comment> \<open>The type T\<close>
   \<comment> \<open>Martin-Löf's book (page 68) discusses elimination and computation.
     Elimination can be derived by computation and equality of types,
     but with an extra premise \<open>C(x)\<close> type \<open>x:T\<close>.
     Also computation can be derived from elimination.\<close>
 
   TF: "T type" and
   TI: "tt : T" and
   TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and
   TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and
   TC: "\<And>p. p : T \<Longrightarrow> p = tt : T"
 
 
 subsection "Tactics and derived rules for Constructive Type Theory"
 
 text \<open>Formation rules.\<close>
 lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
   and formL_rls = ProdFL SumFL PlusFL EqFL
 
 text \<open>
   Introduction rules. OMITTED:
   \<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>.
 \<close>
 lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
   and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
 
 text \<open>
   Elimination rules. OMITTED:
   \<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close>
   \<^item> \<open>TE\<close>, because it does not involve a constructor.
 \<close>
 lemmas elim_rls = NE ProdE SumE PlusE FE
   and elimL_rls = NEL ProdEL SumEL PlusEL FEL
 
 text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close>
 lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
 
 text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close>
 lemmas element_rls = intr_rls elim_rls
 
 text \<open>Definitions are (meta)equality axioms.\<close>
 lemmas basic_defs = fst_def snd_def
 
 text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close>
 lemma SumIL2: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <c,d> = <a,b> : Sum(A,B)"
   by (rule sym_elem) (rule SumIL; rule sym_elem)
 
 lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
 
 text \<open>
   Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>.
   A more natural form of product elimination.
 \<close>
 lemma subst_prodE:
   assumes "p: Prod(A,B)"
     and "a: A"
     and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)"
   shows "c(p`a): C(p`a)"
   by (rule assms ProdE)+
 
 
 subsection \<open>Tactics for type checking\<close>
 
 ML \<open>
 local
 
-fun is_rigid_elem (Const(\<^const_name>\<open>Elem\<close>,_) $ a $ _) = not(is_Var (head_of a))
-  | is_rigid_elem (Const(\<^const_name>\<open>Eqelem\<close>,_) $ a $ _ $ _) = not(is_Var (head_of a))
-  | is_rigid_elem (Const(\<^const_name>\<open>Type\<close>,_) $ a) = not(is_Var (head_of a))
+fun is_rigid_elem \<^Const_>\<open>Elem for a _\<close> = not(is_Var (head_of a))
+  | is_rigid_elem \<^Const_>\<open>Eqelem for a _ _\<close> = not(is_Var (head_of a))
+  | is_rigid_elem \<^Const_>\<open>Type for a\<close> = not(is_Var (head_of a))
   | is_rigid_elem _ = false
 
 in
 
 (*Try solving a:A or a=b:A by assumption provided a is rigid!*)
 fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) =>
   if is_rigid_elem (Logic.strip_assums_concl prem)
   then assume_tac ctxt i else no_tac)
 
 fun ASSUME ctxt tf i = test_assume_tac ctxt i  ORELSE  tf i
 
 end
 \<close>
 
 text \<open>
   For simplification: type formation and checking,
   but no equalities between terms.
 \<close>
 lemmas routine_rls = form_rls formL_rls refl_type element_rls
 
 ML \<open>
 fun routine_tac rls ctxt prems =
   ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls)));
 
 (*Solve all subgoals "A type" using formation rules. *)
 val form_net = Tactic.build_net @{thms form_rls};
 fun form_tac ctxt =
   REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net));
 
 (*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
 fun typechk_tac ctxt thms =
   let val tac =
     filt_resolve_from_net_tac ctxt 3
       (Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls}))
   in  REPEAT_FIRST (ASSUME ctxt tac)  end
 
 (*Solve a:A (a flexible, A rigid) by introduction rules.
   Cannot use stringtrees (filt_resolve_tac) since
   goals like ?a:SUM(A,B) have a trivial head-string *)
 fun intr_tac ctxt thms =
   let val tac =
     filt_resolve_from_net_tac ctxt 1
       (Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls}))
   in  REPEAT_FIRST (ASSUME ctxt tac)  end
 
 (*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
 fun equal_tac ctxt thms =
   REPEAT_FIRST
     (ASSUME ctxt
       (filt_resolve_from_net_tac ctxt 3
         (Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem}))))
 \<close>
 
 method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close>
 method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close>
 method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close>
 method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close>
 
 
 subsection \<open>Simplification\<close>
 
 text \<open>To simplify the type in a goal.\<close>
 lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B"
   apply (rule equal_types)
    apply (rule_tac [2] sym_type)
    apply assumption+
   done
 
 text \<open>Simplify the parameter of a unary type operator.\<close>
 lemma subst_eqtyparg:
   assumes 1: "a=c : A"
     and 2: "\<And>z. z:A \<Longrightarrow> B(z) type"
   shows "B(a) = B(c)"
   apply (rule subst_typeL)
    apply (rule_tac [2] refl_type)
    apply (rule 1)
   apply (erule 2)
   done
 
 text \<open>Simplification rules for Constructive Type Theory.\<close>
 lemmas reduction_rls = comp_rls [THEN trans_elem]
 
 ML \<open>
 (*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification.
   Uses other intro rules to avoid changing flexible goals.*)
 val eqintr_net = Tactic.build_net @{thms EqI intr_rls}
 fun eqintr_tac ctxt =
   REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net))
 
 (** Tactics that instantiate CTT-rules.
     Vars in the given terms will be incremented!
     The (rtac EqE i) lets them apply to equality judgements. **)
 
 fun NE_tac ctxt sp i =
   TRY (resolve_tac ctxt @{thms EqE} i) THEN
   Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i
 
 fun SumE_tac ctxt sp i =
   TRY (resolve_tac ctxt @{thms EqE} i) THEN
   Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i
 
 fun PlusE_tac ctxt sp i =
   TRY (resolve_tac ctxt @{thms EqE} i) THEN
   Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i
 
 (** Predicate logic reasoning, WITH THINNING!!  Procedures adapted from NJ. **)
 
 (*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *)
 fun add_mp_tac ctxt i =
   resolve_tac ctxt @{thms subst_prodE} i  THEN  assume_tac ctxt i  THEN  assume_tac ctxt i
 
 (*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *)
 fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i  THEN  assume_tac ctxt i
 
 (*"safe" when regarded as predicate calculus rules*)
 val safe_brls = sort (make_ord lessb)
     [ (true, @{thm FE}), (true,asm_rl),
       (false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
 
 val unsafe_brls =
     [ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
       (true, @{thm subst_prodE}) ]
 
 (*0 subgoals vs 1 or more*)
 val (safe0_brls, safep_brls) =
     List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
 
 fun safestep_tac ctxt thms i =
     form_tac ctxt ORELSE
     resolve_tac ctxt thms i  ORELSE
     biresolve_tac ctxt safe0_brls i  ORELSE  mp_tac ctxt i  ORELSE
     DETERM (biresolve_tac ctxt safep_brls i)
 
 fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i)
 
 fun step_tac ctxt thms = safestep_tac ctxt thms  ORELSE'  biresolve_tac ctxt unsafe_brls
 
 (*Fails unless it solves the goal!*)
 fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms)
 \<close>
 
 method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close>
 method_setup NE = \<open>
   Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s))
 \<close>
 method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close>
 method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close>
 
 ML_file \<open>rew.ML\<close>
 method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close>
 method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close>
 
 
 subsection \<open>The elimination rules for fst/snd\<close>
 
 lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A"
   apply (unfold basic_defs)
   apply (erule SumE)
   apply assumption
   done
 
 text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close>
 lemma SumE_snd:
   assumes major: "p: Sum(A,B)"
     and "A type"
     and "\<And>x. x:A \<Longrightarrow> B(x) type"
   shows "snd(p) : B(fst(p))"
   apply (unfold basic_defs)
   apply (rule major [THEN SumE])
   apply (rule SumC [THEN subst_eqtyparg, THEN replace_type])
       apply (typechk assms)
   done
 
 
 section \<open>The two-element type (booleans and conditionals)\<close>
 
 definition Bool :: "t"
   where "Bool \<equiv> T+T"
 
 definition true :: "i"
   where "true \<equiv> inl(tt)"
 
 definition false :: "i"
   where "false \<equiv> inr(tt)"
 
 definition cond :: "[i,i,i]\<Rightarrow>i"
   where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)"
 
 lemmas bool_defs = Bool_def true_def false_def cond_def
 
 
 subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close>
 
 text \<open>Formation rule.\<close>
 lemma boolF: "Bool type"
   unfolding bool_defs by typechk
 
 text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close>
 
 lemma boolI_true: "true : Bool"
   unfolding bool_defs by typechk
 
 lemma boolI_false: "false : Bool"
   unfolding bool_defs by typechk
 
 text \<open>Elimination rule: typing of \<open>cond\<close>.\<close>
 lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)"
   unfolding bool_defs
   apply (typechk; erule TE)
    apply typechk
   done
 
 lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk>
   \<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)"
   unfolding bool_defs
   apply (rule PlusEL)
     apply (erule asm_rl refl_elem [THEN TEL])+
   done
 
 text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close>
 
 lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)"
   unfolding bool_defs
   apply (rule comp_rls)
     apply typechk
    apply (erule_tac [!] TE)
    apply typechk
   done
 
 lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)"
   unfolding bool_defs
   apply (rule comp_rls)
     apply typechk
    apply (erule_tac [!] TE)
    apply typechk
   done
 
 section \<open>Elementary arithmetic\<close>
 
 subsection \<open>Arithmetic operators and their definitions\<close>
 
 definition add :: "[i,i]\<Rightarrow>i"   (infixr "#+" 65)
   where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))"
 
 definition diff :: "[i,i]\<Rightarrow>i"   (infixr "-" 65)
   where "a-b \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))"
 
 definition absdiff :: "[i,i]\<Rightarrow>i"   (infixr "|-|" 65)
   where "a|-|b \<equiv> (a-b) #+ (b-a)"
 
 definition mult :: "[i,i]\<Rightarrow>i"   (infixr "#*" 70)
   where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)"
 
 definition mod :: "[i,i]\<Rightarrow>i"   (infixr "mod" 70)
   where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v) |-| b, 0, \<lambda>x y. succ(v)))"
 
 definition div :: "[i,i]\<Rightarrow>i"   (infixr "div" 70)
   where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))"
 
 lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def
 
 
 subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close>
 
 subsubsection \<open>Addition\<close>
 
 text \<open>Typing of \<open>add\<close>: short and long versions.\<close>
 
 lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N"
   unfolding arith_defs by typechk
 
 lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N"
   unfolding arith_defs by equal
 
 
 text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close>
 
 lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N"
   unfolding arith_defs by rew
 
 lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N"
   unfolding arith_defs by rew
 
 
 subsubsection \<open>Multiplication\<close>
 
 text \<open>Typing of \<open>mult\<close>: short and long versions.\<close>
 
 lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N"
   unfolding arith_defs by (typechk add_typing)
 
 lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N"
   unfolding arith_defs by (equal add_typingL)
 
 
 text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close>
 
 lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N"
   unfolding arith_defs by rew
 
 lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N"
   unfolding arith_defs by rew
 
 
 subsubsection \<open>Difference\<close>
 
 text \<open>Typing of difference.\<close>
 
 lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N"
   unfolding arith_defs by typechk
 
 lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N"
   unfolding arith_defs by equal
 
 
 text \<open>Computation for difference: 0 and successor cases.\<close>
 
 lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N"
   unfolding arith_defs by rew
 
 text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close>
 
 lemma diff_0_eq_0: "b:N \<Longrightarrow> 0 - b = 0 : N"
   unfolding arith_defs
   apply (NE b)
     apply hyp_rew
   done
 
 text \<open>
   Essential to simplify FIRST!!  (Else we get a critical pair)
   \<open>succ(a) - succ(b)\<close> rewrites to \<open>pred(succ(a) - b)\<close>.
 \<close>
 lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) - succ(b) = a - b : N"
   unfolding arith_defs
   apply hyp_rew
   apply (NE b)
     apply hyp_rew
   done
 
 
 subsection \<open>Simplification\<close>
 
 lemmas arith_typing_rls = add_typing mult_typing diff_typing
   and arith_congr_rls = add_typingL mult_typingL diff_typingL
 
 lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls
 
 lemmas arithC_rls =
   addC0 addC_succ
   multC0 multC_succ
   diffC0 diff_0_eq_0 diff_succ_succ
 
 ML \<open>
   structure Arith_simp = TSimpFun(
     val refl = @{thm refl_elem}
     val sym = @{thm sym_elem}
     val trans = @{thm trans_elem}
     val refl_red = @{thm refl_red}
     val trans_red = @{thm trans_red}
     val red_if_equal = @{thm red_if_equal}
     val default_rls = @{thms arithC_rls comp_rls}
     val routine_tac = routine_tac @{thms arith_typing_rls routine_rls}
   )
 
   fun arith_rew_tac ctxt prems =
     make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems))
 
   fun hyp_arith_rew_tac ctxt prems =
     make_rew_tac ctxt
       (Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems))
 \<close>
 
 method_setup arith_rew = \<open>
   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths))
 \<close>
 
 method_setup hyp_arith_rew = \<open>
   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths))
 \<close>
 
 
 subsection \<open>Addition\<close>
 
 text \<open>Associative law for addition.\<close>
 lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N"
   apply (NE a)
     apply hyp_arith_rew
   done
 
 text \<open>Commutative law for addition.  Can be proved using three inductions.
   Must simplify after first induction!  Orientation of rewrites is delicate.\<close>
 lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N"
   apply (NE a)
     apply hyp_arith_rew
    apply (rule sym_elem)
    prefer 2
    apply (NE b)
      prefer 4
      apply (NE b)
        apply hyp_arith_rew
   done
 
 
 subsection \<open>Multiplication\<close>
 
 text \<open>Right annihilation in product.\<close>
 lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N"
   apply (NE a)
     apply hyp_arith_rew
   done
 
 text \<open>Right successor law for multiplication.\<close>
 lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N"
   apply (NE a)
     apply (hyp_arith_rew add_assoc [THEN sym_elem])
   apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+
   done
 
 text \<open>Commutative law for multiplication.\<close>
 lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N"
   apply (NE a)
     apply (hyp_arith_rew mult_0_right mult_succ_right)
   done
 
 text \<open>Addition distributes over multiplication.\<close>
 lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
   apply (NE a)
     apply (hyp_arith_rew add_assoc [THEN sym_elem])
   done
 
 text \<open>Associative law for multiplication.\<close>
 lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N"
   apply (NE a)
     apply (hyp_arith_rew add_mult_distrib)
   done
 
 
 subsection \<open>Difference\<close>
 
 text \<open>
   Difference on natural numbers, without negative numbers
   \<^item> \<open>a - b = 0\<close>  iff  \<open>a \<le> b\<close>
   \<^item> \<open>a - b = succ(c)\<close> iff \<open>a > b\<close>
 \<close>
 
 lemma diff_self_eq_0: "a:N \<Longrightarrow> a - a = 0 : N"
   apply (NE a)
     apply hyp_arith_rew
   done
 
 
 lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> c #+ 0 = c : N"
   by (rule addC0 [THEN [3] add_commute [THEN trans_elem]])
 
 text \<open>
   Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x - b) = x\<close>.
   An example of induction over a quantified formula (a product).
   Uses rewriting with a quantified, implicative inductive hypothesis.
 \<close>
 schematic_goal add_diff_inverse_lemma:
   "b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, b-x, 0) \<longrightarrow> Eq(N, b #+ (x-b), x)"
   apply (NE b)
     \<comment> \<open>strip one "universal quantifier" but not the "implication"\<close>
     apply (rule_tac [3] intr_rls)
     \<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x - succ(u)) = x\<close>\<close>
      prefer 4
      apply (NE x)
        apply assumption
     \<comment> \<open>Prepare for simplification of types -- the antecedent \<open>succ(u) \<le> x\<close>\<close>
       apply (rule_tac [2] replace_type)
        apply (rule_tac [1] replace_type)
         apply arith_rew
     \<comment> \<open>Solves first 0 goal, simplifies others.  Two sugbgoals remain.
     Both follow by rewriting, (2) using quantified induction hyp.\<close>
    apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
     apply (hyp_arith_rew add_0_right)
   apply assumption
   done
 
 text \<open>
   Version of above with premise \<open>b - a = 0\<close> i.e. \<open>a \<ge> b\<close>.
   Using @{thm ProdE} does not work -- for \<open>?B(?a)\<close> is ambiguous.
   Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme;
   the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically.
 \<close>
 lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b - a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (a-b) = a : N"
   apply (rule EqE)
   apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
     apply (assumption | rule EqI)+
   done
 
 
 subsection \<open>Absolute difference\<close>
 
 text \<open>Typing of absolute difference: short and long versions.\<close>
 
 lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N"
   unfolding arith_defs by typechk
 
 lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N"
   unfolding arith_defs by equal
 
 lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N"
   unfolding absdiff_def by (arith_rew diff_self_eq_0)
 
 lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N"
   unfolding absdiff_def by hyp_arith_rew
 
 lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) |-| succ(b)  =  a |-| b : N"
   unfolding absdiff_def by hyp_arith_rew
 
 text \<open>Note how easy using commutative laws can be?  ...not always...\<close>
 lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b = b |-| a : N"
   unfolding absdiff_def
   apply (rule add_commute)
    apply (typechk diff_typing)
   done
 
 text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close>
 schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)"
   apply (NE a)
     apply (rule_tac [3] replace_type)
      apply arith_rew
   apply intr  \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
    apply (rule_tac [2] zero_ne_succ [THEN FE])
      apply (erule_tac [3] EqE [THEN sym_elem])
     apply (typechk add_typing)
   done
 
 text \<open>
   Version of above with the premise \<open>a + b = 0\<close>.
   Again, resolution instantiates variables in @{thm ProdE}.
 \<close>
 lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N"
   apply (rule EqE)
   apply (rule add_eq0_lemma [THEN ProdE])
     apply (rule_tac [3] EqI)
     apply typechk
   done
 
 text \<open>Here is a lemma to infer \<open>a - b = 0\<close> and \<open>b - a = 0\<close> from \<open>a |-| b = 0\<close>, below.\<close>
 schematic_goal absdiff_eq0_lem:
   "\<lbrakk>a:N; b:N; a |-| b = 0 : N\<rbrakk> \<Longrightarrow> ?a : Eq(N, a-b, 0) \<times> Eq(N, b-a, 0)"
   apply (unfold absdiff_def)
   apply intr
    apply eqintr
    apply (rule_tac [2] add_eq0)
      apply (rule add_eq0)
        apply (rule_tac [6] add_commute [THEN trans_elem])
          apply (typechk diff_typing)
   done
 
 text \<open>If \<open>a |-| b = 0\<close> then \<open>a = b\<close>
   proof: \<open>a - b = 0\<close> and \<open>b - a = 0\<close>, so \<open>b = a + (b - a) = a + 0 = a\<close>.
 \<close>
 lemma absdiff_eq0: "\<lbrakk>a |-| b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N"
   apply (rule EqE)
   apply (rule absdiff_eq0_lem [THEN SumE])
      apply eqintr
   apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem])
      apply (erule_tac [3] EqE)
     apply (hyp_arith_rew add_0_right)
   done
 
 
 subsection \<open>Remainder and Quotient\<close>
 
 text \<open>Typing of remainder: short and long versions.\<close>
 
 lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N"
   unfolding mod_def by (typechk absdiff_typing)
 
 lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N"
   unfolding mod_def by (equal absdiff_typingL)
 
 
 text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close>
 
 lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N"
   unfolding mod_def by (rew absdiff_typing)
 
 lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
   succ(a) mod b = rec(succ(a mod b) |-| b, 0, \<lambda>x y. succ(a mod b)) : N"
   unfolding mod_def by (rew absdiff_typing)
 
 
 text \<open>Typing of quotient: short and long versions.\<close>
 
 lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N"
   unfolding div_def by (typechk absdiff_typing mod_typing)
 
 lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N"
   unfolding div_def by (equal absdiff_typingL mod_typingL)
 
 lemmas div_typing_rls = mod_typing div_typing absdiff_typing
 
 
 text \<open>Computation for quotient: 0 and successor cases.\<close>
 
 lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N"
   unfolding div_def by (rew mod_typing absdiff_typing)
 
 lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
   succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N"
   unfolding div_def by (rew mod_typing)
 
 
 text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close>
 lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
   succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), \<lambda>x y. a div b) : N"
   apply (rule divC_succ [THEN trans_elem])
     apply (rew div_typing_rls modC_succ)
   apply (NE "succ (a mod b) |-|b")
     apply (rew mod_typing div_typing absdiff_typing)
   done
 
 text \<open>For case analysis on whether a number is 0 or a successor.\<close>
 lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) :
   Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))"
   apply (NE a)
     apply (rule_tac [3] PlusI_inr)
      apply (rule_tac [2] PlusI_inl)
       apply eqintr
      apply equal
   done
 
 text \<open>Main Result. Holds when \<open>b\<close> is 0 since \<open>a mod 0 = a\<close> and \<open>a div 0 = 0\<close>.\<close>
 lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N"
   apply (NE a)
     apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
   apply (rule EqE)
     \<comment> \<open>case analysis on \<open>succ(u mod b) |-| b\<close>\<close>
   apply (rule_tac a1 = "succ (u mod b) |-| b" in iszero_decidable [THEN PlusE])
     apply (erule_tac [3] SumE)
     apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
     \<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close>
   apply (rule add_typingL [THEN trans_elem])
     apply (erule EqE [THEN absdiff_eq0, THEN sym_elem])
      apply (rule_tac [3] refl_elem)
      apply (hyp_arith_rew div_typing_rls)
   done
 
 end