58385

1 
(* Title: HOL/Datatype_Examples/Misc_N2M.thy


2 
Author: Dmitriy Traytel, TU Muenchen


3 
Copyright 2014


4 


5 
Miscellaneous datatype definitions.


6 
*)


7 


8 
header \<open>Miscellaneous Tests for the NestedtoMutual Reduction\<close>


9 


10 
theory Misc_N2M


11 
imports "~~/src/HOL/Library/BNF_Axiomatization"


12 
begin


13 


14 
locale misc


15 
begin


16 


17 
datatype 'a li = Ni  Co 'a "'a li"


18 
datatype 'a tr = Tr "'a \<Rightarrow> 'a tr li"


19 


20 
primrec (nonexhaustive)


21 
f_tl :: "'a \<Rightarrow> 'a tr li \<Rightarrow> nat" and


22 
f_t :: "'a \<Rightarrow> 'a tr \<Rightarrow> nat"


23 
where


24 
"f_tl _ Ni = 0" 


25 
"f_t a (Tr ts) = 1 + f_tl a (ts a)"


26 


27 
datatype ('a, 'b) l = N  C 'a 'b "('a, 'b) l"


28 
datatype ('a, 'b) t = T "(('a, 'b) t, 'a) l" "(('a, 'b) t, 'b) l"


29 


30 
primrec (nonexhaustive)


31 
f_tl2 :: "(('a, 'a) t, 'a) l \<Rightarrow> nat" and


32 
f_t2 :: "('a, 'a) t \<Rightarrow> nat"


33 
where


34 
"f_tl2 N = 0" 


35 
"f_t2 (T ts us) = f_tl2 ts + f_tl2 us"


36 


37 
primrec (nonexhaustive)


38 
g_tla :: "(('a, 'b) t, 'a) l \<Rightarrow> nat" and


39 
g_tlb :: "(('a, 'b) t, 'b) l \<Rightarrow> nat" and


40 
g_t :: "('a, 'b) t \<Rightarrow> nat"


41 
where


42 
"g_tla N = 0" 


43 
"g_tlb N = 1" 


44 
"g_t (T ts us) = g_tla ts + g_tlb us"


45 


46 


47 
datatype


48 
'a l1 = N1  C1 'a "'a l1"


49 


50 
datatype


51 
('a, 'b) t1 = T1 'a 'b "('a, 'b) t1 l1" "(nat \<times> ('a, 'b) t1) l1" and


52 
('a, 'b) t2 = T2 "('a, 'b) t1"


53 


54 
primrec


55 
h1_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


56 
h1_natl1 :: "(nat \<times> (nat, 'a) t1) l1 \<Rightarrow> nat" and


57 
h1_t1 :: "(nat, 'a) t1 \<Rightarrow> nat"


58 
where


59 
"h1_tl1 N1 = 0" 


60 
"h1_tl1 (C1 t ts) = h1_t1 t + h1_tl1 ts" 


61 
"h1_natl1 N1 = Suc 0" 


62 
"h1_natl1 (C1 n ts) = fst n + h1_natl1 ts" 


63 
"h1_t1 (T1 n _ ts _) = n + h1_tl1 ts"


64 


65 
end


66 


67 


68 
bnf_axiomatization ('a, 'b) M0 [wits: "('a, 'b) M0"]


69 
bnf_axiomatization ('a, 'b) N0 [wits: "('a, 'b) N0"]


70 
bnf_axiomatization ('a, 'b) K0 [wits: "('a, 'b) K0"]


71 
bnf_axiomatization ('a, 'b) L0 [wits: "('a, 'b) L0"]


72 
bnf_axiomatization ('a, 'b, 'c) G0 [wits: "('a, 'b, 'c) G0"]


73 


74 
locale (*co*)mplicated


75 
begin


76 


77 
datatype 'a M = CM "('a, 'a M) M0"


78 
datatype 'a N = CN "('a N, 'a) N0"


79 
datatype ('a, 'b) K = CK "('a, ('a, 'b) L) K0"


80 
and ('a, 'b) L = CL "('b, ('a, 'b) K) L0"


81 
datatype 'a G = CG "('a, ('a G, 'a G N) K, ('a G M, 'a G) L) G0"


82 


83 
primrec


84 
fG :: "'a G \<Rightarrow> 'a G"


85 
and fK :: "('a G, 'a G N) K \<Rightarrow> ('a G, 'a G N) K"


86 
and fL :: "('a G, 'a G N) L \<Rightarrow> ('a G, 'a G N) L"


87 
and fM :: "'a G M \<Rightarrow> 'a G M" where


88 
"fG (CG x) = CG (map_G0 id fK (map_L fM fG) x)"


89 
 "fK (CK y) = CK (map_K0 fG fL y)"


90 
 "fL (CL z) = CL (map_L0 (map_N fG) fK z)"


91 
 "fM (CM w) = CM (map_M0 fG fM w)"


92 
thm fG_def fK_def fL_def fM_def fG.simps fK.simps fL.simps fM.simps


93 


94 
end


95 


96 
locale complicated


97 
begin


98 


99 
codatatype 'a M = CM "('a, 'a M) M0"


100 
codatatype 'a N = CN "('a N, 'a) N0"


101 
codatatype ('a, 'b) K = CK "('a, ('a, 'b) L) K0"


102 
and ('a, 'b) L = CL "('b, ('a, 'b) K) L0"


103 
codatatype 'a G = CG "('a, ('a G, 'a G N) K, ('a G M, 'a G) L) G0"


104 


105 
primcorec


106 
fG :: "'a G \<Rightarrow> 'a G"


107 
and fK :: "('a G, 'a G N) K \<Rightarrow> ('a G, 'a G N) K"


108 
and fL :: "('a G, 'a G N) L \<Rightarrow> ('a G, 'a G N) L"


109 
and fM :: "'a G M \<Rightarrow> 'a G M" where


110 
"fG x = CG (map_G0 id fK (map_L fM fG) (un_CG x))"


111 
 "fK y = CK (map_K0 fG fL (un_CK y))"


112 
 "fL z = CL (map_L0 (map_N fG) fK (un_CL z))"


113 
 "fM w = CM (map_M0 fG fM (un_CM w))"


114 
thm fG_def fK_def fL_def fM_def fG.simps fK.simps fL.simps fM.simps


115 


116 
end


117 


118 
datatype ('a, 'b) F1 = F1 'a 'b


119 
datatype F2 = F2 "((unit, nat) F1, F2) F1"  F2'


120 
datatype 'a kk1 = K1 'a  K2 "'a kk2" and 'a kk2 = K3 "'a kk1"


121 
datatype 'a ll1 = L1 'a  L2 "'a ll2 kk2" and 'a ll2 = L3 "'a ll1"


122 


123 
datatype_compat F1


124 
datatype_compat F2


125 
datatype_compat kk1 kk2


126 
datatype_compat ll1 ll2


127 


128 


129 


130 
section \<open>Deep Nesting\<close>


131 


132 
datatype 'a tree = Empty  Node 'a "'a tree list"


133 
datatype_compat tree


134 


135 
datatype 'a ttree = TEmpty  TNode 'a "'a ttree list tree"


136 
datatype_compat ttree


137 


138 
datatype 'a tttree = TEmpty  TNode 'a "'a tttree list ttree list tree"


139 
datatype_compat tttree


140 
(*


141 
datatype 'a ttttree = TEmpty  TNode 'a "'a ttttree list tttree list ttree list tree"


142 
datatype_compat ttttree


143 
*)


144 


145 
datatype ('a,'b)complex =


146 
C1 nat "'a ttree"


147 
 C2 "('a,'b)complex list tree tree" 'b "('a,'b)complex" "('a,'b)complex2 ttree list"


148 
and ('a,'b)complex2 = D1 "('a,'b)complex ttree"


149 
datatype_compat complex complex2


150 


151 
datatype 'a F = F1 'a  F2 "'a F"


152 
datatype 'a G = G1 'a  G2 "'a G F"


153 
datatype H = H1  H2 "H G"


154 


155 
datatype_compat F


156 
datatype_compat G


157 
datatype_compat H


158 


159 
section \<open>Primrec cache\<close>


160 


161 
datatype 'a l = N  C 'a "'a l"


162 
datatype ('a, 'b) t = T 'a 'b "('a, 'b) t l"


163 


164 
context linorder


165 
begin


166 


167 
(* slow *)


168 
primrec


169 
f1_tl :: "(nat, 'a) t l \<Rightarrow> nat" and


170 
f1_t :: "(nat, 'a) t \<Rightarrow> nat"


171 
where


172 
"f1_tl N = 0" 


173 
"f1_tl (C t ts) = f1_t t + f1_tl ts" 


174 
"f1_t (T n _ ts) = n + f1_tl ts"


175 


176 
(* should be fast *)


177 
primrec


178 
f2_t :: "(nat, 'b) t \<Rightarrow> nat" and


179 
f2_tl :: "(nat, 'b) t l \<Rightarrow> nat"


180 
where


181 
"f2_tl N = 0" 


182 
"f2_tl (C t ts) = f2_t t + f2_tl ts" 


183 
"f2_t (T n _ ts) = n + f2_tl ts"


184 


185 
end


186 


187 
(* should be fast *)


188 
primrec


189 
g1_t :: "('a, int) t \<Rightarrow> nat" and


190 
g1_tl :: "('a, int) t l \<Rightarrow> nat"


191 
where


192 
"g1_t (T _ _ ts) = g1_tl ts" 


193 
"g1_tl N = 0" 


194 
"g1_tl (C _ ts) = g1_tl ts"


195 


196 
(* should be fast *)


197 
primrec


198 
g2_t :: "(int, int) t \<Rightarrow> nat" and


199 
g2_tl :: "(int, int) t l \<Rightarrow> nat"


200 
where


201 
"g2_t (T _ _ ts) = g2_tl ts" 


202 
"g2_tl N = 0" 


203 
"g2_tl (C _ ts) = g2_tl ts"


204 


205 


206 
datatype


207 
'a l1 = N1  C1 'a "'a l2" and


208 
'a l2 = N2  C2 'a "'a l1"


209 


210 
primrec


211 
sum_l1 :: "'a::{zero,plus} l1 \<Rightarrow> 'a" and


212 
sum_l2 :: "'a::{zero,plus} l2 \<Rightarrow> 'a"


213 
where


214 
"sum_l1 N1 = 0" 


215 
"sum_l1 (C1 n ns) = n + sum_l2 ns" 


216 
"sum_l2 N2 = 0" 


217 
"sum_l2 (C2 n ns) = n + sum_l1 ns"


218 


219 
datatype


220 
('a, 'b) t1 = T1 'a 'b "('a, 'b) t1 l1" and


221 
('a, 'b) t2 = T2 "('a, 'b) t1"


222 


223 
(* slow *)


224 
primrec


225 
h1_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


226 
h1_tl2 :: "(nat, 'a) t1 l2 \<Rightarrow> nat" and


227 
h1_t1 :: "(nat, 'a) t1 \<Rightarrow> nat"


228 
where


229 
"h1_tl1 N1 = 0" 


230 
"h1_tl1 (C1 t ts) = h1_t1 t + h1_tl2 ts" 


231 
"h1_tl2 N2 = 0" 


232 
"h1_tl2 (C2 t ts) = h1_t1 t + h1_tl1 ts" 


233 
"h1_t1 (T1 n _ ts) = n + h1_tl1 ts"


234 


235 
(* should be fast *)


236 
primrec


237 
h2_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


238 
h2_tl2 :: "(nat, 'a) t1 l2 \<Rightarrow> nat" and


239 
h2_t1 :: "(nat, 'a) t1 \<Rightarrow> nat"


240 
where


241 
"h2_tl1 N1 = 0" 


242 
"h2_tl1 (C1 t ts) = h2_t1 t + h2_tl2 ts" 


243 
"h2_tl2 N2 = 0" 


244 
"h2_tl2 (C2 t ts) = h2_t1 t + h2_tl1 ts" 


245 
"h2_t1 (T1 n _ ts) = n + h2_tl1 ts"


246 


247 
(* should be fast *)


248 
primrec


249 
h3_tl2 :: "(nat, 'a) t1 l2 \<Rightarrow> nat" and


250 
h3_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


251 
h3_t1 :: "(nat, 'a) t1 \<Rightarrow> nat"


252 
where


253 
"h3_tl1 N1 = 0" 


254 
"h3_tl1 (C1 t ts) = h3_t1 t + h3_tl2 ts" 


255 
"h3_tl2 N2 = 0" 


256 
"h3_tl2 (C2 t ts) = h3_t1 t + h3_tl1 ts" 


257 
"h3_t1 (T1 n _ ts) = n + h3_tl1 ts"


258 


259 
(* should be fast *)


260 
primrec


261 
i1_tl2 :: "(nat, 'a) t1 l2 \<Rightarrow> nat" and


262 
i1_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


263 
i1_t1 :: "(nat, 'a) t1 \<Rightarrow> nat" and


264 
i1_t2 :: "(nat, 'a) t2 \<Rightarrow> nat"


265 
where


266 
"i1_tl1 N1 = 0" 


267 
"i1_tl1 (C1 t ts) = i1_t1 t + i1_tl2 ts" 


268 
"i1_tl2 N2 = 0" 


269 
"i1_tl2 (C2 t ts) = i1_t1 t + i1_tl1 ts" 


270 
"i1_t1 (T1 n _ ts) = n + i1_tl1 ts" 


271 
"i1_t2 (T2 t) = i1_t1 t"


272 


273 
(* should be fast *)


274 
primrec


275 
j1_t2 :: "(nat, 'a) t2 \<Rightarrow> nat" and


276 
j1_t1 :: "(nat, 'a) t1 \<Rightarrow> nat" and


277 
j1_tl1 :: "(nat, 'a) t1 l1 \<Rightarrow> nat" and


278 
j1_tl2 :: "(nat, 'a) t1 l2 \<Rightarrow> nat"


279 
where


280 
"j1_tl1 N1 = 0" 


281 
"j1_tl1 (C1 t ts) = j1_t1 t + j1_tl2 ts" 


282 
"j1_tl2 N2 = 0" 


283 
"j1_tl2 (C2 t ts) = j1_t1 t + j1_tl1 ts" 


284 
"j1_t1 (T1 n _ ts) = n + j1_tl1 ts" 


285 
"j1_t2 (T2 t) = j1_t1 t"


286 


287 


288 
datatype 'a l3 = N3  C3 'a "'a l3"


289 
datatype 'a l4 = N4  C4 'a "'a l4"


290 
datatype ('a, 'b) t3 = T3 'a 'b "('a, 'b) t3 l3" "('a, 'b) t3 l4"


291 


292 
(* slow *)


293 
primrec


294 
k1_tl3 :: "(nat, 'a) t3 l3 \<Rightarrow> nat" and


295 
k1_tl4 :: "(nat, 'a) t3 l4 \<Rightarrow> nat" and


296 
k1_t3 :: "(nat, 'a) t3 \<Rightarrow> nat"


297 
where


298 
"k1_tl3 N3 = 0" 


299 
"k1_tl3 (C3 t ts) = k1_t3 t + k1_tl3 ts" 


300 
"k1_tl4 N4 = 0" 


301 
"k1_tl4 (C4 t ts) = k1_t3 t + k1_tl4 ts" 


302 
"k1_t3 (T3 n _ ts us) = n + k1_tl3 ts + k1_tl4 us"


303 


304 
(* should be fast *)


305 
primrec


306 
k2_tl4 :: "(nat, int) t3 l4 \<Rightarrow> nat" and


307 
k2_tl3 :: "(nat, int) t3 l3 \<Rightarrow> nat" and


308 
k2_t3 :: "(nat, int) t3 \<Rightarrow> nat"


309 
where


310 
"k2_tl4 N4 = 0" 


311 
"k2_tl4 (C4 t ts) = k2_t3 t + k2_tl4 ts" 


312 
"k2_tl3 N3 = 0" 


313 
"k2_tl3 (C3 t ts) = k2_t3 t + k2_tl3 ts" 


314 
"k2_t3 (T3 n _ ts us) = n + k2_tl3 ts + k2_tl4 us"


315 


316 


317 
datatype ('a, 'b) l5 = N5  C5 'a 'b "('a, 'b) l5"


318 
datatype ('a, 'b) l6 = N6  C6 'a 'b "('a, 'b) l6"


319 
datatype ('a, 'b, 'c) t4 = T4 'a 'b "(('a, 'b, 'c) t4, 'b) l5" "(('a, 'b, 'c) t4, 'c) l6"


320 


321 
(* slow *)


322 
primrec


323 
l1_tl5 :: "((nat, 'a, 'b) t4, 'a) l5 \<Rightarrow> nat" and


324 
l1_tl6 :: "((nat, 'a, 'b) t4, 'b) l6 \<Rightarrow> nat" and


325 
l1_t4 :: "(nat, 'a, 'b) t4 \<Rightarrow> nat"


326 
where


327 
"l1_tl5 N5 = 0" 


328 
"l1_tl5 (C5 t _ ts) = l1_t4 t + l1_tl5 ts" 


329 
"l1_tl6 N6 = 0" 


330 
"l1_tl6 (C6 t _ ts) = l1_t4 t + l1_tl6 ts" 


331 
"l1_t4 (T4 n _ ts us) = n + l1_tl5 ts + l1_tl6 us"


332 


333 


334 


335 
section \<open>Primcorec Cache\<close>


336 


337 
codatatype 'a col = N  C 'a "'a col"


338 
codatatype ('a, 'b) cot = T 'a 'b "('a, 'b) cot col"


339 


340 
context linorder


341 
begin


342 


343 
(* slow *)


344 
primcorec


345 
f1_cotcol :: "nat \<Rightarrow> (nat, 'a) cot col" and


346 
f1_cot :: "nat \<Rightarrow> (nat, 'a) cot"


347 
where


348 
"n = 0 \<Longrightarrow> f1_cotcol n = N" 


349 
"_ \<Longrightarrow> f1_cotcol n = C (f1_cot n) (f1_cotcol n)" 


350 
"f1_cot n = T n undefined (f1_cotcol n)"


351 


352 
(* should be fast *)


353 
primcorec


354 
f2_cotcol :: "nat \<Rightarrow> (nat, 'b) cot col" and


355 
f2_cot :: "nat \<Rightarrow> (nat, 'b) cot"


356 
where


357 
"n = 0 \<Longrightarrow> f2_cotcol n = N" 


358 
"_ \<Longrightarrow> f2_cotcol n = C (f2_cot n) (f2_cotcol n)" 


359 
"f2_cot n = T n undefined (f2_cotcol n)"


360 


361 
end


362 


363 
(* should be fast *)


364 
primcorec


365 
g1_cot :: "int \<Rightarrow> (int, 'a) cot" and


366 
g1_cotcol :: "int \<Rightarrow> (int, 'a) cot col"


367 
where


368 
"g1_cot n = T n undefined (g1_cotcol n)" 


369 
"n = 0 \<Longrightarrow> g1_cotcol n = N" 


370 
"_ \<Longrightarrow> g1_cotcol n = C (g1_cot n) (g1_cotcol n)"


371 


372 
(* should be fast *)


373 
primcorec


374 
g2_cot :: "int \<Rightarrow> (int, int) cot" and


375 
g2_cotcol :: "int \<Rightarrow> (int, int) cot col"


376 
where


377 
"g2_cot n = T n undefined (g2_cotcol n)" 


378 
"n = 0 \<Longrightarrow> g2_cotcol n = N" 


379 
"_ \<Longrightarrow> g2_cotcol n = C (g2_cot n) (g2_cotcol n)"


380 


381 


382 
codatatype


383 
'a col1 = N1  C1 'a "'a col2" and


384 
'a col2 = N2  C2 'a "'a col1"


385 


386 
codatatype


387 
('a, 'b) cot1 = T1 'a 'b "('a, 'b) cot1 col1" and


388 
('a, 'b) cot2 = T2 "('a, 'b) cot1"


389 


390 
(* slow *)


391 
primcorec


392 
h1_cotcol1 :: "nat \<Rightarrow> (nat, 'a) cot1 col1" and


393 
h1_cotcol2 :: "nat \<Rightarrow> (nat, 'a) cot1 col2" and


394 
h1_cot1 :: "nat \<Rightarrow> (nat, 'a) cot1"


395 
where


396 
"h1_cotcol1 n = C1 (h1_cot1 n) (h1_cotcol2 n)" 


397 
"h1_cotcol2 n = C2 (h1_cot1 n) (h1_cotcol1 n)" 


398 
"h1_cot1 n = T1 n undefined (h1_cotcol1 n)"


399 


400 
(* should be fast *)


401 
primcorec


402 
h2_cotcol1 :: "nat \<Rightarrow> (nat, 'a) cot1 col1" and


403 
h2_cotcol2 :: "nat \<Rightarrow> (nat, 'a) cot1 col2" and


404 
h2_cot1 :: "nat \<Rightarrow> (nat, 'a) cot1"


405 
where


406 
"h2_cotcol1 n = C1 (h2_cot1 n) (h2_cotcol2 n)" 


407 
"h2_cotcol2 n = C2 (h2_cot1 n) (h2_cotcol1 n)" 


408 
"h2_cot1 n = T1 n undefined (h2_cotcol1 n)"


409 


410 
(* should be fast *)


411 
primcorec


412 
h3_cotcol2 :: "nat \<Rightarrow> (nat, 'a) cot1 col2" and


413 
h3_cotcol1 :: "nat \<Rightarrow> (nat, 'a) cot1 col1" and


414 
h3_cot1 :: "nat \<Rightarrow> (nat, 'a) cot1"


415 
where


416 
"h3_cotcol1 n = C1 (h3_cot1 n) (h3_cotcol2 n)" 


417 
"h3_cotcol2 n = C2 (h3_cot1 n) (h3_cotcol1 n)" 


418 
"h3_cot1 n = T1 n undefined (h3_cotcol1 n)"


419 


420 
(* should be fast *)


421 
primcorec


422 
i1_cotcol2 :: "nat \<Rightarrow> (nat, 'a) cot1 col2" and


423 
i1_cotcol1 :: "nat \<Rightarrow> (nat, 'a) cot1 col1" and


424 
i1_cot1 :: "nat \<Rightarrow> (nat, 'a) cot1" and


425 
i1_cot2 :: "nat \<Rightarrow> (nat, 'a) cot2"


426 
where


427 
"i1_cotcol1 n = C1 (i1_cot1 n) (i1_cotcol2 n)" 


428 
"i1_cotcol2 n = C2 (i1_cot1 n) (i1_cotcol1 n)" 


429 
"i1_cot1 n = T1 n undefined (i1_cotcol1 n)" 


430 
"i1_cot2 n = T2 (i1_cot1 n)"


431 


432 
(* should be fast *)


433 
primcorec


434 
j1_cot2 :: "nat \<Rightarrow> (nat, 'a) cot2" and


435 
j1_cot1 :: "nat \<Rightarrow> (nat, 'a) cot1" and


436 
j1_cotcol1 :: "nat \<Rightarrow> (nat, 'a) cot1 col1" and


437 
j1_cotcol2 :: "nat \<Rightarrow> (nat, 'a) cot1 col2"


438 
where


439 
"j1_cotcol1 n = C1 (j1_cot1 n) (j1_cotcol2 n)" 


440 
"j1_cotcol2 n = C2 (j1_cot1 n) (j1_cotcol1 n)" 


441 
"j1_cot1 n = T1 n undefined (j1_cotcol1 n)" 


442 
"j1_cot2 n = T2 (j1_cot1 n)"


443 


444 


445 
codatatype 'a col3 = N3  C3 'a "'a col3"


446 
codatatype 'a col4 = N4  C4 'a "'a col4"


447 
codatatype ('a, 'b) cot3 = T3 'a 'b "('a, 'b) cot3 col3" "('a, 'b) cot3 col4"


448 


449 
(* slow *)


450 
primcorec


451 
k1_cotcol3 :: "nat \<Rightarrow> (nat, 'a) cot3 col3" and


452 
k1_cotcol4 :: "nat \<Rightarrow> (nat, 'a) cot3 col4" and


453 
k1_cot3 :: "nat \<Rightarrow> (nat, 'a) cot3"


454 
where


455 
"k1_cotcol3 n = C3 (k1_cot3 n) (k1_cotcol3 n)" 


456 
"k1_cotcol4 n = C4 (k1_cot3 n) (k1_cotcol4 n)" 


457 
"k1_cot3 n = T3 n undefined (k1_cotcol3 n) (k1_cotcol4 n)"


458 


459 
(* should be fast *)


460 
primcorec


461 
k2_cotcol4 :: "nat \<Rightarrow> (nat, 'a) cot3 col4" and


462 
k2_cotcol3 :: "nat \<Rightarrow> (nat, 'a) cot3 col3" and


463 
k2_cot3 :: "nat \<Rightarrow> (nat, 'a) cot3"


464 
where


465 
"k2_cotcol4 n = C4 (k2_cot3 n) (k2_cotcol4 n)" 


466 
"k2_cotcol3 n = C3 (k2_cot3 n) (k2_cotcol3 n)" 


467 
"k2_cot3 n = T3 n undefined (k2_cotcol3 n) (k2_cotcol4 n)"


468 


469 
end
