# Theory Wfd

```(*  Title:      CCL/Wfd.thy
Author:     Martin Coen, Cambridge University Computer Laboratory
*)

section ‹Well-founded relations in CCL›

theory Wfd
imports Trancl Type Hered
begin

definition Wfd :: "[i set] ⇒ o"
where "Wfd(R) == ALL P.(ALL x.(ALL y.<y,x> : R ⟶ y:P) ⟶ x:P) ⟶ (ALL a. a:P)"

definition wf :: "[i set] ⇒ i set"
where "wf(R) == {x. x:R ∧ Wfd(R)}"

definition wmap :: "[i⇒i,i set] ⇒ i set"
where "wmap(f,R) == {p. EX x y. p=<x,y> ∧ <f(x),f(y)> : R}"

definition lex :: "[i set,i set] => i set"      (infixl "**" 70)
where "ra**rb == {p. EX a a' b b'. p = <<a,b>,<a',b'>> ∧ (<a,a'> : ra | (a=a' ∧ <b,b'> : rb))}"

definition NatPR :: "i set"
where "NatPR == {p. EX x:Nat. p=<x,succ(x)>}"

definition ListPR :: "i set ⇒ i set"
where "ListPR(A) == {p. EX h:A. EX t:List(A). p=<t,h\$t>}"

lemma wfd_induct:
assumes 1: "Wfd(R)"
and 2: "⋀x. ALL y. <y,x>: R ⟶ P(y) ⟹ P(x)"
shows "P(a)"
apply (rule 1 [unfolded Wfd_def, rule_format, THEN CollectD])
using 2 apply blast
done

lemma wfd_strengthen_lemma:
assumes 1: "⋀x y.<x,y> : R ⟹ Q(x)"
and 2: "ALL x. (ALL y. <y,x> : R ⟶ y : P) ⟶ x : P"
and 3: "⋀x. Q(x) ⟹ x:P"
shows "a:P"
apply (rule 2 [rule_format])
using 1 3
apply blast
done

method_setup wfd_strengthen = ‹
Scan.lift Parse.embedded_inner_syntax >> (fn s => fn ctxt =>
SIMPLE_METHOD' (fn i =>
Rule_Insts.res_inst_tac ctxt [((("Q", 0), Position.none), s)] [] @{thm wfd_strengthen_lemma} i
THEN assume_tac ctxt (i + 1)))
›

lemma wf_anti_sym: "⟦Wfd(r); <a,x>:r; <x,a>:r⟧ ⟹ P"
apply (subgoal_tac "ALL x. <a,x>:r ⟶ <x,a>:r ⟶ P")
apply blast
apply (erule wfd_induct)
apply blast
done

lemma wf_anti_refl: "⟦Wfd(r); <a,a>: r⟧ ⟹ P"
apply (rule wf_anti_sym)
apply assumption+
done

subsection ‹Irreflexive transitive closure›

lemma trancl_wf:
assumes 1: "Wfd(R)"
shows "Wfd(R^+)"
apply (unfold Wfd_def)
apply (rule allI ballI impI)+
(*must retain the universal formula for later use!*)
apply (rule allE, assumption)
apply (erule mp)
apply (rule 1 [THEN wfd_induct])
apply (rule impI [THEN allI])
apply (erule tranclE)
apply blast
apply (erule spec [THEN mp, THEN spec, THEN mp])
apply assumption+
done

subsection ‹Lexicographic Ordering›

lemma lexXH:
"p : ra**rb ⟷ (EX a a' b b'. p = <<a,b>,<a',b'>> ∧ (<a,a'> : ra | a=a' ∧ <b,b'> : rb))"
unfolding lex_def by blast

lemma lexI1: "<a,a'> : ra ⟹ <<a,b>,<a',b'>> : ra**rb"
by (blast intro!: lexXH [THEN iffD2])

lemma lexI2: "<b,b'> : rb ⟹ <<a,b>,<a,b'>> : ra**rb"
by (blast intro!: lexXH [THEN iffD2])

lemma lexE:
assumes 1: "p : ra**rb"
and 2: "⋀a a' b b'. ⟦<a,a'> : ra; p=<<a,b>,<a',b'>>⟧ ⟹ R"
and 3: "⋀a b b'. ⟦<b,b'> : rb; p = <<a,b>,<a,b'>>⟧ ⟹ R"
shows R
apply (rule 1 [THEN lexXH [THEN iffD1], THEN exE])
using 2 3
apply blast
done

lemma lex_pair: "⟦p : r**s; ⋀a a' b b'. p = <<a,b>,<a',b'>> ⟹ P⟧ ⟹P"
apply (erule lexE)
apply blast+
done

lemma lex_wf:
assumes 1: "Wfd(R)"
and 2: "Wfd(S)"
shows "Wfd(R**S)"
apply (unfold Wfd_def)
apply safe
apply (wfd_strengthen "λx. EX a b. x=<a,b>")
apply (blast elim!: lex_pair)
apply (subgoal_tac "ALL a b.<a,b>:P")
apply blast
apply (rule 1 [THEN wfd_induct, THEN allI])
apply (rule 2 [THEN wfd_induct, THEN allI]) back
apply (fast elim!: lexE)
done

subsection ‹Mapping›

lemma wmapXH: "p : wmap(f,r) ⟷ (EX x y. p=<x,y> ∧ <f(x),f(y)> : r)"
unfolding wmap_def by blast

lemma wmapI: "<f(a),f(b)> : r ⟹ <a,b> : wmap(f,r)"
by (blast intro!: wmapXH [THEN iffD2])

lemma wmapE: "⟦p : wmap(f,r); ⋀a b. ⟦<f(a),f(b)> : r; p=<a,b>⟧ ⟹ R⟧ ⟹ R"
by (blast dest!: wmapXH [THEN iffD1])

lemma wmap_wf:
assumes 1: "Wfd(r)"
shows "Wfd(wmap(f,r))"
apply (unfold Wfd_def)
apply clarify
apply (subgoal_tac "ALL b. ALL a. f (a) = b ⟶ a:P")
apply blast
apply (rule 1 [THEN wfd_induct, THEN allI])
apply clarify
apply (erule spec [THEN mp])
apply (safe elim!: wmapE)
apply (erule spec [THEN mp, THEN spec, THEN mp])
apply assumption
apply (rule refl)
done

subsection ‹Projections›

lemma wfstI: "<xa,ya> : r ⟹ <<xa,xb>,<ya,yb>> : wmap(fst,r)"
apply (rule wmapI)
apply simp
done

lemma wsndI: "<xb,yb> : r ⟹ <<xa,xb>,<ya,yb>> : wmap(snd,r)"
apply (rule wmapI)
apply simp
done

lemma wthdI: "<xc,yc> : r ⟹ <<xa,<xb,xc>>,<ya,<yb,yc>>> : wmap(thd,r)"
apply (rule wmapI)
apply simp
done

subsection ‹Ground well-founded relations›

lemma wfI: "⟦Wfd(r);  a : r⟧ ⟹ a : wf(r)"
unfolding wf_def by blast

lemma Empty_wf: "Wfd({})"
unfolding Wfd_def by (blast elim: EmptyXH [THEN iffD1, THEN FalseE])

lemma wf_wf: "Wfd(wf(R))"
unfolding wf_def
apply (rule_tac Q = "Wfd(R)" in excluded_middle [THEN disjE])
apply simp_all
apply (rule Empty_wf)
done

lemma NatPRXH: "p : NatPR ⟷ (EX x:Nat. p=<x,succ(x)>)"
unfolding NatPR_def by blast

lemma ListPRXH: "p : ListPR(A) ⟷ (EX h:A. EX t:List(A).p=<t,h\$t>)"
unfolding ListPR_def by blast

lemma NatPRI: "x : Nat ⟹ <x,succ(x)> : NatPR"
by (auto simp: NatPRXH)

lemma ListPRI: "⟦t : List(A); h : A⟧ ⟹ <t,h \$ t> : ListPR(A)"
by (auto simp: ListPRXH)

lemma NatPR_wf: "Wfd(NatPR)"
apply (unfold Wfd_def)
apply clarify
apply (wfd_strengthen "λx. x:Nat")
apply (fastforce iff: NatPRXH)
apply (erule Nat_ind)
apply (fastforce iff: NatPRXH)+
done

lemma ListPR_wf: "Wfd(ListPR(A))"
apply (unfold Wfd_def)
apply clarify
apply (wfd_strengthen "λx. x:List (A)")
apply (fastforce iff: ListPRXH)
apply (erule List_ind)
apply (fastforce iff: ListPRXH)+
done

subsection ‹General Recursive Functions›

lemma letrecT:
assumes 1: "a : A"
and 2: "⋀p g. ⟦p:A; ALL x:{x: A. <x,p>:wf(R)}. g(x) : D(x)⟧ ⟹ h(p,g) : D(p)"
shows "letrec g x be h(x,g) in g(a) : D(a)"
apply (rule 1 [THEN rev_mp])
apply (rule wf_wf [THEN wfd_induct])
apply (subst letrecB)
apply (rule impI)
apply (erule 2)
apply blast
done

lemma SPLITB: "SPLIT(<a,b>,B) = B(a,b)"
unfolding SPLIT_def
apply (rule set_ext)
apply blast
done

lemma letrec2T:
assumes "a : A"
and "b : B"
and "⋀p q g. ⟦p:A; q:B;
ALL x:A. ALL y:{y: B. <<x,y>,<p,q>>:wf(R)}. g(x,y) : D(x,y)⟧ ⟹
h(p,q,g) : D(p,q)"
shows "letrec g x y be h(x,y,g) in g(a,b) : D(a,b)"
apply (unfold letrec2_def)
apply (rule SPLITB [THEN subst])
apply (assumption | rule letrecT pairT splitT assms)+
apply (subst SPLITB)
apply (assumption | rule ballI SubtypeI assms)+
apply (rule SPLITB [THEN subst])
apply (assumption | rule letrecT SubtypeI pairT splitT assms |
erule bspec SubtypeE sym [THEN subst])+
done

lemma lem: "SPLIT(<a,<b,c>>,λx xs. SPLIT(xs,λy z. B(x,y,z))) = B(a,b,c)"

lemma letrec3T:
assumes "a : A"
and "b : B"
and "c : C"
and "⋀p q r g. ⟦p:A; q:B; r:C;
ALL x:A. ALL y:B. ALL z:{z:C. <<x,<y,z>>,<p,<q,r>>> : wf(R)}.
g(x,y,z) : D(x,y,z) ⟧ ⟹
h(p,q,r,g) : D(p,q,r)"
shows "letrec g x y z be h(x,y,z,g) in g(a,b,c) : D(a,b,c)"
apply (unfold letrec3_def)
apply (rule lem [THEN subst])
apply (assumption | rule letrecT pairT splitT assms)+
apply (assumption | rule ballI SubtypeI assms)+
apply (rule lem [THEN subst])
apply (assumption | rule letrecT SubtypeI pairT splitT assms |
erule bspec SubtypeE sym [THEN subst])+
done

lemmas letrecTs = letrecT letrec2T letrec3T

subsection ‹Type Checking for Recursive Calls›

lemma rcallT:
"⟦ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x);
g(a) : D(a) ⟹ g(a) : E;  a:A;  <a,p>:wf(R)⟧ ⟹ g(a) : E"
by blast

lemma rcall2T:
"⟦ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y);
g(a,b) : D(a,b) ⟹ g(a,b) : E; a:A; b:B; <<a,b>,<p,q>>:wf(R)⟧ ⟹ g(a,b) : E"
by blast

lemma rcall3T:
"⟦ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}. g(x,y,z):D(x,y,z);
g(a,b,c) : D(a,b,c) ⟹ g(a,b,c) : E;
a:A; b:B; c:C; <<a,<b,c>>,<p,<q,r>>> : wf(R)⟧ ⟹ g(a,b,c) : E"
by blast

lemmas rcallTs = rcallT rcall2T rcall3T

subsection ‹Instantiating an induction hypothesis with an equality assumption›

lemma hyprcallT:
assumes 1: "g(a) = b"
and 2: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x)"
and 3: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ⟹ b=g(a) ⟹ g(a) : D(a) ⟹ P"
and 4: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ⟹ a:A"
and 5: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ⟹ <a,p>:wf(R)"
shows P
apply (rule 3 [OF 2, OF 1 [symmetric]])
apply (rule rcallT [OF 2])
apply assumption
apply (rule 4 [OF 2])
apply (rule 5 [OF 2])
done

lemma hyprcall2T:
assumes 1: "g(a,b) = c"
and 2: "ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y)"
and 3: "⟦c = g(a,b); g(a,b) : D(a,b)⟧ ⟹ P"
and 4: "a:A"
and 5: "b:B"
and 6: "<<a,b>,<p,q>>:wf(R)"
shows P
apply (rule 3)
apply (rule 1 [symmetric])
apply (rule rcall2T)
apply (rule 2)
apply assumption
apply (rule 4)
apply (rule 5)
apply (rule 6)
done

lemma hyprcall3T:
assumes 1: "g(a,b,c) = d"
and 2: "ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z)"
and 3: "⟦d = g(a,b,c); g(a,b,c) : D(a,b,c)⟧ ⟹ P"
and 4: "a:A"
and 5: "b:B"
and 6: "c:C"
and 7: "<<a,<b,c>>,<p,<q,r>>> : wf(R)"
shows P
apply (rule 3)
apply (rule 1 [symmetric])
apply (rule rcall3T)
apply (rule 2)
apply assumption
apply (rule 4)
apply (rule 5)
apply (rule 6)
apply (rule 7)
done

lemmas hyprcallTs = hyprcallT hyprcall2T hyprcall3T

subsection ‹Rules to Remove Induction Hypotheses after Type Checking›

lemma rmIH1: "⟦ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x); P⟧ ⟹ P" .

lemma rmIH2: "⟦ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y); P⟧ ⟹ P" .

lemma rmIH3:
"⟦ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z); P⟧ ⟹ P" .

lemmas rmIHs = rmIH1 rmIH2 rmIH3

subsection ‹Lemmas for constructors and subtypes›

(* 0-ary constructors do not need additional rules as they are handled *)
(*                                      correctly by applying SubtypeI *)

lemma Subtype_canTs:
"⋀a b A B P. a : {x:A. b:{y:B(a).P(<x,y>)}} ⟹ <a,b> : {x:Sigma(A,B).P(x)}"
"⋀a A B P. a : {x:A. P(inl(x))} ⟹ inl(a) : {x:A+B. P(x)}"
"⋀b A B P. b : {x:B. P(inr(x))} ⟹ inr(b) : {x:A+B. P(x)}"
"⋀a P. a : {x:Nat. P(succ(x))} ⟹ succ(a) : {x:Nat. P(x)}"
"⋀h t A P. h : {x:A. t : {y:List(A).P(x\$y)}} ⟹ h\$t : {x:List(A).P(x)}"
by (assumption | rule SubtypeI canTs icanTs | erule SubtypeE)+

lemma letT: "⟦f(t):B; ¬t=bot⟧ ⟹ let x be t in f(x) : B"
apply (erule letB [THEN ssubst])
apply assumption
done

lemma applyT2: "⟦a:A; f : Pi(A,B)⟧ ⟹ f ` a  : B(a)"
apply (erule applyT)
apply assumption
done

lemma rcall_lemma1: "⟦a:A; a:A ⟹ P(a)⟧ ⟹ a : {x:A. P(x)}"
by blast

lemma rcall_lemma2: "⟦a:{x:A. Q(x)}; ⟦a:A; Q(a)⟧ ⟹ P(a)⟧ ⟹ a : {x:A. P(x)}"
by blast

lemmas rcall_lemmas = asm_rl rcall_lemma1 SubtypeD1 rcall_lemma2

subsection ‹Typechecking›

ML ‹
local

val type_rls =
@{thms canTs} @ @{thms icanTs} @ @{thms applyT2} @ @{thms ncanTs} @ @{thms incanTs} @
@{thms precTs} @ @{thms letrecTs} @ @{thms letT} @ @{thms Subtype_canTs};

fun bvars \<^Const_>‹Pure.all _ for ‹Abs(s,_,t)›› l = bvars t (s::l)
| bvars _ l = l

fun get_bno l n \<^Const_>‹Pure.all _ for ‹Abs(s,_,t)›› = get_bno (s::l) n t
| get_bno l n \<^Const_>‹Trueprop for t› = get_bno l n t
| get_bno l n \<^Const_>‹Ball _ for _ ‹Abs(s,_,t)›› = get_bno (s::l) (n+1) t
| get_bno l n \<^Const_>‹mem _ for t _› = get_bno l n t
| get_bno l n (t \$ s) = get_bno l n t
| get_bno l n (Bound m) = (m-length(l),n)

(* Not a great way of identifying induction hypothesis! *)
fun could_IH x = Term.could_unify(x,hd (Thm.prems_of @{thm rcallT})) orelse
Term.could_unify(x,hd (Thm.prems_of @{thm rcall2T})) orelse
Term.could_unify(x,hd (Thm.prems_of @{thm rcall3T}))

fun IHinst tac rls = SUBGOAL (fn (Bi,i) =>
let val bvs = bvars Bi []
val ihs = filter could_IH (Logic.strip_assums_hyp Bi)
val rnames = map (fn x =>
let val (a,b) = get_bno [] 0 x
in (nth bvs a, b) end) ihs
fun try_IHs [] = no_tac
| try_IHs ((x,y)::xs) =
tac [((("g", 0), Position.none), x)] (nth rls (y - 1)) i ORELSE (try_IHs xs)
in try_IHs rnames end)

fun is_rigid_prog t =
(case (Logic.strip_assums_concl t) of
\<^Const_>‹Trueprop for \<^Const_>‹mem _ for a _›› => null (Term.add_vars a [])
| _ => false)

in

fun rcall_tac ctxt i =
let fun tac ps rl i = Rule_Insts.res_inst_tac ctxt ps [] rl i THEN assume_tac ctxt i
in IHinst tac @{thms rcallTs} i end
THEN eresolve_tac ctxt @{thms rcall_lemmas} i

fun raw_step_tac ctxt prems i =
assume_tac ctxt i ORELSE
resolve_tac ctxt (prems @ type_rls) i ORELSE
rcall_tac ctxt i ORELSE
ematch_tac ctxt @{thms SubtypeE} i ORELSE
match_tac ctxt @{thms SubtypeI} i

fun tc_step_tac ctxt prems = SUBGOAL (fn (Bi,i) =>
if is_rigid_prog Bi then raw_step_tac ctxt prems i else no_tac)

fun typechk_tac ctxt rls i = SELECT_GOAL (REPEAT_FIRST (tc_step_tac ctxt rls)) i

(*** Clean up Correctness Condictions ***)

fun clean_ccs_tac ctxt =
let fun tac ps rl i = Rule_Insts.eres_inst_tac ctxt ps [] rl i THEN assume_tac ctxt i in
TRY (REPEAT_FIRST (IHinst tac @{thms hyprcallTs} ORELSE'
eresolve_tac ctxt ([asm_rl, @{thm SubtypeE}] @ @{thms rmIHs}) ORELSE'
hyp_subst_tac ctxt))
end

fun gen_ccs_tac ctxt rls i =
SELECT_GOAL (REPEAT_FIRST (tc_step_tac ctxt rls) THEN clean_ccs_tac ctxt) i

end
›

method_setup typechk = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (typechk_tac ctxt ths))
›

method_setup clean_ccs = ‹
Scan.succeed (SIMPLE_METHOD o clean_ccs_tac)
›

method_setup gen_ccs = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (gen_ccs_tac ctxt ths))
›

subsection ‹Evaluation›

named_theorems eval "evaluation rules"

ML ‹
fun eval_tac ths =
Subgoal.FOCUS_PREMS (fn {context = ctxt, prems, ...} =>
let val eval_rules = Named_Theorems.get ctxt \<^named_theorems>‹eval›
in DEPTH_SOLVE_1 (resolve_tac ctxt (ths @ prems @ rev eval_rules) 1) end)
›

method_setup eval = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (CHANGED o eval_tac ths ctxt))
›

lemmas eval_rls [eval] = trueV falseV pairV lamV caseVtrue caseVfalse caseVpair caseVlam

lemma applyV [eval]:
assumes "f ⤏ lam x. b(x)"
and "b(a) ⤏ c"
shows "f ` a ⤏ c"
unfolding apply_def by (eval assms)

lemma letV:
assumes 1: "t ⤏ a"
and 2: "f(a) ⤏ c"
shows "let x be t in f(x) ⤏ c"
apply (unfold let_def)
apply (rule 1 [THEN canonical])
apply (tactic ‹
REPEAT (DEPTH_SOLVE_1 (resolve_tac \<^context> (@{thms assms} @ @{thms eval_rls}) 1 ORELSE
eresolve_tac \<^context> @{thms substitute} 1))›)
done

lemma fixV: "f(fix(f)) ⤏ c ⟹ fix(f) ⤏ c"
apply (unfold fix_def)
apply (rule applyV)
apply (rule lamV)
apply assumption
done

lemma letrecV:
"h(t,λy. letrec g x be h(x,g) in g(y)) ⤏ c ⟹
letrec g x be h(x,g) in g(t) ⤏ c"
apply (unfold letrec_def)
apply (assumption | rule fixV applyV  lamV)+
done

lemmas [eval] = letV letrecV fixV

lemma V_rls [eval]:
"true ⤏ true"
"false ⤏ false"
"⋀b c t u. ⟦b⤏true; t⤏c⟧ ⟹ if b then t else u ⤏ c"
"⋀b c t u. ⟦b⤏false; u⤏c⟧ ⟹ if b then t else u ⤏ c"
"⋀a b. <a,b> ⤏ <a,b>"
"⋀a b c t h. ⟦t ⤏ <a,b>; h(a,b) ⤏ c⟧ ⟹ split(t,h) ⤏ c"
"zero ⤏ zero"
"⋀n. succ(n) ⤏ succ(n)"
"⋀c n t u. ⟦n ⤏ zero; t ⤏ c⟧ ⟹ ncase(n,t,u) ⤏ c"
"⋀c n t u x. ⟦n ⤏ succ(x); u(x) ⤏ c⟧ ⟹ ncase(n,t,u) ⤏ c"
"⋀c n t u. ⟦n ⤏ zero; t ⤏ c⟧ ⟹ nrec(n,t,u) ⤏ c"
"⋀c n t u x. ⟦n⤏succ(x); u(x,nrec(x,t,u))⤏c⟧ ⟹ nrec(n,t,u)⤏c"
"[] ⤏ []"
"⋀h t. h\$t ⤏ h\$t"
"⋀c l t u. ⟦l ⤏ []; t ⤏ c⟧ ⟹ lcase(l,t,u) ⤏ c"
"⋀c l t u x xs. ⟦l ⤏ x\$xs; u(x,xs) ⤏ c⟧ ⟹ lcase(l,t,u) ⤏ c"
"⋀c l t u. ⟦l ⤏ []; t ⤏ c⟧ ⟹ lrec(l,t,u) ⤏ c"
"⋀c l t u x xs. ⟦l⤏x\$xs; u(x,xs,lrec(xs,t,u))⤏c⟧ ⟹ lrec(l,t,u)⤏c"
unfolding data_defs by eval+

subsection ‹Factorial›

schematic_goal
"letrec f n be ncase(n,succ(zero),λx. nrec(n,zero,λy g. nrec(f(x),g,λz h. succ(h))))
in f(succ(succ(zero))) ⤏ ?a"
by eval

schematic_goal
"letrec f n be ncase(n,succ(zero),λx. nrec(n,zero,λy g. nrec(f(x),g,λz h. succ(h))))
in f(succ(succ(succ(zero)))) ⤏ ?a"
by eval

subsection ‹Less Than Or Equal›

schematic_goal
"letrec f p be split(p,λm n. ncase(m,true,λx. ncase(n,false,λy. f(<x,y>))))
in f(<succ(zero), succ(zero)>) ⤏ ?a"
by eval

schematic_goal
"letrec f p be split(p,λm n. ncase(m,true,λx. ncase(n,false,λy. f(<x,y>))))
in f(<succ(zero), succ(succ(succ(succ(zero))))>) ⤏ ?a"
by eval

schematic_goal
"letrec f p be split(p,λm n. ncase(m,true,λx. ncase(n,false,λy. f(<x,y>))))
in f(<succ(succ(succ(succ(succ(zero))))), succ(succ(succ(succ(zero))))>) ⤏ ?a"
by eval

subsection ‹Reverse›

schematic_goal
"letrec id l be lcase(l,[],λx xs. x\$id(xs))
in id(zero\$succ(zero)\$[]) ⤏ ?a"
by eval

schematic_goal
"letrec rev l be lcase(l,[],λx xs. lrec(rev(xs),x\$[],λy ys g. y\$g))
in rev(zero\$succ(zero)\$(succ((lam x. x)`succ(zero)))\$([])) ⤏ ?a"
by eval

end
```