%   ORIGINAL: 'h4/thm/list/LIST_REL_def_e3'
% 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: !bs b as a R. 'h4/const/list/LIST_REL' R ('h4/const/list/CONS' a as) ('h4/const/list/CONS' b bs) <=> R a b /\ 'h4/const/list/LIST_REL' R as bs
%   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: !bs b as a R. 'h4/const/list/LIST_REL' R ('h4/const/list/CONS' a as) ('h4/const/list/CONS' b bs) <=> happ (happ R a) b /\ 'h4/const/list/LIST_REL' R as bs
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_3f5714,V_3f5710]: ![F, G]: (![X]: s(V_3f5710,happ(s(fun(V_3f5714,V_3f5710),F),s(V_3f5714,X))) = s(V_3f5710,happ(s(fun(V_3f5714,V_3f5710),G),s(V_3f5714,X))) => s(fun(V_3f5714,V_3f5710),F) = s(fun(V_3f5714,V_3f5710),G))).
fof('h4/thm/list/LIST_REL_def_e3', conjecture, ![A,B]: ![Bs, B0, As, A0, R]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),R),s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,A0),s('h4/type/list/list'(A),As))),s('h4/type/list/list'(B),'h4/const/list/CONS'(s(B,B0),s('h4/type/list/list'(B),Bs)))))) <=> (p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),R),s(A,A0))),s(B,B0)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),R),s('h4/type/list/list'(A),As),s('h4/type/list/list'(B),Bs))))))).
