%   ORIGINAL: 'h4/thm/list/LIST_REL_cases_'
% 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: !R a0 a1. 'h4/const/list/LIST_REL' R a0 a1 <=> a0 = 'h4/const/list/NIL' /\ a1 = 'h4/const/list/NIL' \/ (?h1 h2 t1 t2. a0 = 'h4/const/list/CONS' h1 t1 /\ a1 = 'h4/const/list/CONS' h2 t2 /\ R h1 h2 /\ 'h4/const/list/LIST_REL' R t1 t2)
%   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: !R a0 a1. 'h4/const/list/LIST_REL' R a0 a1 <=> a0 = 'h4/const/list/NIL' /\ a1 = 'h4/const/list/NIL' \/ (?h1 h2 t1 t2. a0 = 'h4/const/list/CONS' h1 t1 /\ a1 = 'h4/const/list/CONS' h2 t2 /\ happ (happ R h1) h2 /\ 'h4/const/list/LIST_REL' R t1 t2)
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_3f5589,V_3f5585]: ![F, G]: (![X]: s(V_3f5585,happ(s(fun(V_3f5589,V_3f5585),F),s(V_3f5589,X))) = s(V_3f5585,happ(s(fun(V_3f5589,V_3f5585),G),s(V_3f5589,X))) => s(fun(V_3f5589,V_3f5585),F) = s(fun(V_3f5589,V_3f5585),G))).
fof('h4/thm/list/LIST_REL_cases_', conjecture, ![A,B]: ![R, A0, A1]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),R),s('h4/type/list/list'(A),A0),s('h4/type/list/list'(B),A1)))) <=> ((s('h4/type/list/list'(A),A0) = s('h4/type/list/list'(A),'h4/const/list/NIL') & s('h4/type/list/list'(B),A1) = s('h4/type/list/list'(B),'h4/const/list/NIL')) | ?[H1, H2, T1, T2]: (s('h4/type/list/list'(A),A0) = s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,H1),s('h4/type/list/list'(A),T1))) & (s('h4/type/list/list'(B),A1) = s('h4/type/list/list'(B),'h4/const/list/CONS'(s(B,H2),s('h4/type/list/list'(B),T2))) & (p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),R),s(A,H1))),s(B,H2)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),R),s('h4/type/list/list'(A),T1),s('h4/type/list/list'(B),T2)))))))))).
