abbreviation Times :: "[t,t]\t" (infixr \\\ 50) where"A \ B \ \_: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 judgments. \<close> axiomatization where
refl_red: "\a. Reduce[a,a]" and
red_if_equal: "\a b A. a = b : A \ Reduce[a,b]" and
trans_red: "\a b c A. \a = b : A; Reduce[b,c]\ \ a = c : A" and
\<comment> \<open>Reflexivity\<close>
refl_type: "\A. A type \ A = A" and
refl_elem: "\a A. a : A \ a = a : A" and
\<comment> \<open>Symmetry\<close>
sym_type: "\A B. A = B \ B = A" and
sym_elem: "\a b A. a = b : A \ b = a : A" and
\<comment> \<open>Transitivity\<close>
trans_type: "\A B C. \A = B; B = C\ \ A = C" and
trans_elem: "\a b c A. \a = b : A; b = c : A\ \ a = c : A" and
equal_types: "\a A B. \a : A; A = B\ \ a : B" and
equal_typesL: "\a b A B. \a = b : A; A = B\ \ a = b : B" and
\<comment> \<open>Substitution\<close>
subst_type: "\a A B. \a : A; \z. z:A \ B(z) type\ \ B(a) type" and
subst_typeL: "\a c A B D. \a = c : A; \z. z:A \ B(z) = D(z)\ \ B(a) = D(c)" and
subst_elem: "\a b A B. \a : A; \z. z:A \ b(z):B(z)\ \ b(a):B(a)" and
subst_elemL: "\a b c d A B. \a = c : A; \z. z:A \ b(z)=d(z) : B(z)\ \ 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: "\a. a : N \ succ(a) : N" and
NI_succL: "\a b. a = b : N \ succ(a) = succ(b) : N" and
NE: "\p a b C. \p: N; a: C(0); \u v. \u: N; v: C(u)\ \ b(u,v): C(succ(u))\ \<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and
NEL: "\p q a b c d C. \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: "\a b C. \a: C(0); \u v. \u: N; v: C(u)\ \ b(u,v): C(succ(u))\ \<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and
NC_succ: "\p a b C. \p: N; a: C(0); \u v. \u: N; v: C(u)\ \ b(u,v): C(succ(u))\ \
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: "\a. \a: N; 0 = succ(a) : N\ \ 0: F" and
\<comment> \<open>The Product of a family of types\<close>
ProdF: "\A B. \A type; \x. x:A \ B(x) type\ \ \x:A. B(x) type" and
ProdFL: "\A B C D. \A = C; \x. x:A \ B(x) = D(x)\ \ \x:A. B(x) = \x:C. D(x)" and
ProdI: "\b A B. \A type; \x. x:A \ b(x):B(x)\ \ \<^bold>\x. b(x) : \x:A. B(x)" and
ProdIL: "\b c A B. \A type; \x. x:A \ b(x) = c(x) : B(x)\ \ \<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and
ProdE: "\p a A B. \p : \x:A. B(x); a : A\ \ p`a : B(a)" and
ProdEL: "\p q a b A B. \p = q: \x:A. B(x); a = b : A\ \ p`a = q`b : B(a)" and
ProdC: "\a b A B. \a : A; \x. x:A \ b(x) : B(x)\ \ (\<^bold>\x. b(x)) ` a = b(a) : B(a)" and
ProdC2: "\p A B. p : \x:A. B(x) \ (\<^bold>\x. p`x) = p : \x:A. B(x)" and
\<comment> \<open>The Sum of a family of types\<close>
SumF: "\A B. \A type; \x. x:A \ B(x) type\ \ \x:A. B(x) type" and
SumFL: "\A B C D. \A = C; \x. x:A \ B(x) = D(x)\ \ \x:A. B(x) = \x:C. D(x)" and
SumI: "\a b A B. \a : A; b : B(a)\ \ : \x:A. B(x)" and
SumIL: "\a b c d A B. \ a = c : A; b = d : B(a)\ \ = : \x:A. B(x)" and
SumE: "\p c A B C. \p: \x:A. B(x); \x y. \x:A; y:B(x)\ \ c(x,y): C()\ \<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and
SumEL: "\p q c d A B C. \p = q : \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: "\a b c A B C. \a: A; b: B(a); \x y. \x:A; y:B(x)\ \ c(x,y): C()\ \<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and
fst_def: "\a. fst(a) \ split(a, \x y. x)" and
snd_def: "\a. snd(a) \ split(a, \x y. y)" and
\<comment> \<open>The sum of two types\<close>
PlusF: "\A B. \A type; B type\ \ A+B type" and
PlusFL: "\A B C D. \A = C; B = D\ \ A+B = C+D" and
PlusI_inl: "\a A B. \a : A; B type\ \ inl(a) : A+B" and
PlusI_inlL: "\a c A B. \a = c : A; B type\ \ inl(a) = inl(c) : A+B" and
PlusI_inr: "\b A B. \A type; b : B\ \ inr(b) : A+B" and
PlusI_inrL: "\b d A B. \A type; b = d : B\ \ inr(b) = inr(d) : A+B" and
PlusE: "\p c d A B C. \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: "\p q c d e f A B C. \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: "\a c d A B C. \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: "\b c d A B C. \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: "\a b A. \A type; a : A; b : A\ \ Eq(A,a,b) type" and
EqFL: "\a b c d A B. \A = B; a = c : A; b = d : A\ \ Eq(A,a,b) = Eq(B,c,d)" and
EqI: "\a b A. a = b : A \ eq : Eq(A,a,b)" and
EqE: "\p a b A. p : Eq(A,a,b) \ 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: "\p a b A. p : Eq(A,a,b) \ p = eq : Eq(A,a,b)" and
\<comment> \<open>The type \<open>F\<close>\<close>
FF: "F type"and
FE: "\p C. \p: F; C type\ \ contr(p) : C" and
FEL: "\p q C. \p = q : F; C type\ \ 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: "\p c C. \p : T; c : C(tt)\ \ c : C(p)" and
TEL: "\p q c d C. \p = q : T; c = d : C(tt)\ \ c = d : C(p)" and
TC: "\p. p : T \ p = tt : T"
subsection "Tactics and derived rules for Constructive Type Theory"
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 judgment.\<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: "\c = a : A; d = b : B(a)\ \ = : Sum(A,B)" by (rule sym_elem) (rule SumIL; rule sym_elem)
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"\z. z: B(a) \ 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_>\<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)
funASSUME 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 (Bires.filt_resolve_from_net_tac ctxt 4 (Bires.build_net (prems @ rls)));
(*Solve all subgoals "A type" using formation rules. *)
val form_net = Bires.build_net @{thms form_rls}; fun form_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (Bires.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 =
Bires.filt_resolve_from_net_tac ctxt 3
(Bires.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 =
Bires.filt_resolve_from_net_tac ctxt 1
(Bires.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
(Bires.filt_resolve_from_net_tac ctxt 3
(Bires.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem})))) \<close>
text\<open>To simplify the type in a goal.\<close> lemma replace_type: "\B = A; a : A\ \ 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: "\z. z:A \ 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 = Bires.build_net @{thms EqI intr_rls} fun eqintr_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (Bires.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 judgments. **)
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 Bires.subgoals_ord
[ (true, @{thm FE}), (true,asm_rl),
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
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}
)
text\<open>Associative law for addition.\<close> lemma add_assoc: "\a:N; b:N; c:N\ \ (a #+ b) #+ c = a #+ (b #+ c) : N" by (NE a) hyp_arith_rew
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: "\a:N; b:N\ \ 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 \ a #* 0 = 0 : N" apply (NE a) apply hyp_arith_rew done
text\<open>Right successor law for multiplication.\<close> lemma mult_succ_right: "\a:N; b:N\ \ 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: "\a:N; b:N\ \ 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: "\a:N; b:N; c:N\ \ (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: "\a:N; b:N; c:N\ \ (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 \ a - a = 0 : N" apply (NE a) apply hyp_arith_rew done
lemma add_0_right: "\c : N; 0 : N; c : N\ \ 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 \ ?a : \x:N. Eq(N, b-x, 0) \ 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: "\a:N; b:N; b - a = 0 : N\ \ 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: "\a:N; b:N\ \ a |-| b : N" unfolding arith_defs by typechk
lemma absdiff_typingL: "\a = c:N; b = d:N\ \ a |-| b = c |-| d : N" unfolding arith_defs by equal
lemma absdiff_self_eq_0: "a:N \ a |-| a = 0 : N" unfolding absdiff_def by (arith_rew diff_self_eq_0)
lemma absdiffC0: "a:N \ 0 |-| a = a : N" unfolding absdiff_def by hyp_arith_rew
lemma absdiff_succ_succ: "\a:N; b:N\ \ 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: "\a:N; b:N\ \ a |-| b = b |-| a : N" unfolding absdiff_def by (rule add_commute) (typechk diff_typing)
text\<open>Computation for quotient: 0 and successor cases.\<close>
lemma divC0: "b:N \ 0 div b = 0 : N" unfolding div_def by (rew mod_typing absdiff_typing)
lemma divC_succ: "\a:N; b:N\ \
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: "\a:N; b:N\ \
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 \ rec(a, inl(eq), \ka kb. inr()) :
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: "\a:N; b:N\ \ 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
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