%   ORIGINAL: 'h4/thm/list/FILTER_EQ_CONS_'
% 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: !P l h lr. 'h4/const/list/FILTER' P l = 'h4/const/list/CONS' h lr <=> (?l1 l2. l = 'h4/const/list/APPEND' ('h4/const/list/APPEND' l1 ('h4/const/list/CONS' h 'h4/const/list/NIL')) l2 /\ 'h4/const/list/FILTER' P l1 = 'h4/const/list/NIL' /\ 'h4/const/list/FILTER' P l2 = lr /\ P h)
%   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: !P l h lr. 'h4/const/list/FILTER' P l = 'h4/const/list/CONS' h lr <=> (?l1 l2. l = 'h4/const/list/APPEND' ('h4/const/list/APPEND' l1 ('h4/const/list/CONS' h 'h4/const/list/NIL')) l2 /\ 'h4/const/list/FILTER' P l1 = 'h4/const/list/NIL' /\ 'h4/const/list/FILTER' P l2 = lr /\ happ P h)
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_3f4963,V_3f4959]: ![F, G]: (![X]: s(V_3f4959,happ(s(fun(V_3f4963,V_3f4959),F),s(V_3f4963,X))) = s(V_3f4959,happ(s(fun(V_3f4963,V_3f4959),G),s(V_3f4963,X))) => s(fun(V_3f4963,V_3f4959),F) = s(fun(V_3f4963,V_3f4959),G))).
fof('h4/thm/list/FILTER_EQ_CONS_', conjecture, ![A]: ![P, L, H, Lr]: (s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P),s('h4/type/list/list'(A),L))) = s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,H),s('h4/type/list/list'(A),Lr))) <=> ?[L1, L2]: (s('h4/type/list/list'(A),L) = s('h4/type/list/list'(A),'h4/const/list/APPEND'(s('h4/type/list/list'(A),'h4/const/list/APPEND'(s('h4/type/list/list'(A),L1),s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,H),s('h4/type/list/list'(A),'h4/const/list/NIL'))))),s('h4/type/list/list'(A),L2))) & (s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P),s('h4/type/list/list'(A),L1))) = s('h4/type/list/list'(A),'h4/const/list/NIL') & (s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P),s('h4/type/list/list'(A),L2))) = s('h4/type/list/list'(A),Lr) & p(s(bool,happ(s(fun(A,bool),P),s(A,H))))))))).
