%   ORIGINAL: 'h4/thm/list/splitAtPki_EQN_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !l k P. 'h4/const/list/splitAtPki' P k l = 'h4/const/option/option_CASE' ('h4/const/while/OLEAST' (\i. 'h4/const/prim_rec/<' i ('h4/const/list/LENGTH' l) /\ P i ('h4/const/list/EL' i l))) (k l 'h4/const/list/NIL') (\i. k ('h4/const/list/TAKE' i l) ('h4/const/list/DROP' i l))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_1. (!k l i. happ (happ (happ _1 k) l) i = happ (happ k ('h4/const/list/TAKE' i l)) ('h4/const/list/DROP' i l)) ==> (!_0. (!P l i. happ (happ (happ _0 P) l) i <=> 'h4/const/prim_rec/<' i ('h4/const/list/LENGTH' l) /\ happ (happ P i) ('h4/const/list/EL' i l)) ==> (!l k P. 'h4/const/list/splitAtPki' P k l = 'h4/const/option/option_CASE' ('h4/const/while/OLEAST' (happ (happ _0 P) l)) (happ (happ k l) 'h4/const/list/NIL') (happ (happ _1 k) l)))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f10644,V_3f10640]: ![F, G]: (![X]: s(V_3f10640,happ(s(fun(V_3f10644,V_3f10640),F),s(V_3f10644,X))) = s(V_3f10640,happ(s(fun(V_3f10644,V_3f10640),G),s(V_3f10644,X))) => s(fun(V_3f10644,V_3f10640),F) = s(fun(V_3f10644,V_3f10640),G))).
fof('h4/thm/list/splitAtPki_EQN_', conjecture, ![A,B]: ![V__1]: (![K, L, I]: s(A,happ(s(fun('h4/type/num/num',A),happ(s(fun('h4/type/list/list'(B),fun('h4/type/num/num',A)),happ(s(fun(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),fun('h4/type/list/list'(B),fun('h4/type/num/num',A))),V__1),s(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),K))),s('h4/type/list/list'(B),L))),s('h4/type/num/num',I))) = s(A,happ(s(fun('h4/type/list/list'(B),A),happ(s(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),K),s('h4/type/list/list'(B),'h4/const/list/TAKE'(s('h4/type/num/num',I),s('h4/type/list/list'(B),L))))),s('h4/type/list/list'(B),'h4/const/list/DROP'(s('h4/type/num/num',I),s('h4/type/list/list'(B),L))))) => ![V__0]: (![P, L, I]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/list/list'(B),fun('h4/type/num/num',bool)),happ(s(fun(fun('h4/type/num/num',fun(B,bool)),fun('h4/type/list/list'(B),fun('h4/type/num/num',bool))),V__0),s(fun('h4/type/num/num',fun(B,bool)),P))),s('h4/type/list/list'(B),L))),s('h4/type/num/num',I)))) <=> (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',I),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(B),L)))))) & p(s(bool,happ(s(fun(B,bool),happ(s(fun('h4/type/num/num',fun(B,bool)),P),s('h4/type/num/num',I))),s(B,'h4/const/list/EL'(s('h4/type/num/num',I),s('h4/type/list/list'(B),L)))))))) => ![L, K, P]: s(A,'h4/const/list/splitAtPki'(s(fun('h4/type/num/num',fun(B,bool)),P),s(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),K),s('h4/type/list/list'(B),L))) = s(A,'h4/const/option/option_CASE'(s('h4/type/option/option'('h4/type/num/num'),'h4/const/while/OLEAST'(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/list/list'(B),fun('h4/type/num/num',bool)),happ(s(fun(fun('h4/type/num/num',fun(B,bool)),fun('h4/type/list/list'(B),fun('h4/type/num/num',bool))),V__0),s(fun('h4/type/num/num',fun(B,bool)),P))),s('h4/type/list/list'(B),L))))),s(A,happ(s(fun('h4/type/list/list'(B),A),happ(s(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),K),s('h4/type/list/list'(B),L))),s('h4/type/list/list'(B),'h4/const/list/NIL'))),s(fun('h4/type/num/num',A),happ(s(fun('h4/type/list/list'(B),fun('h4/type/num/num',A)),happ(s(fun(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),fun('h4/type/list/list'(B),fun('h4/type/num/num',A))),V__1),s(fun('h4/type/list/list'(B),fun('h4/type/list/list'(B),A)),K))),s('h4/type/list/list'(B),L)))))))).
