%   ORIGINAL: 'h4/thm/list/LIST_REL_CONJ_'
% 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: !l2 l1 Q P. 'h4/const/list/LIST_REL' (\a b. P a b /\ Q a b) l1 l2 <=> 'h4/const/list/LIST_REL' (\a b. P a b) l1 l2 /\ 'h4/const/list/LIST_REL' (\a b. Q a b) l1 l2
%   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: !_3. (!P a b. happ (happ (happ _3 P) a) b <=> happ (happ P a) b) ==> (!_2. (!P a. happ (happ _2 P) a = happ (happ _3 P) a) ==> (!_1. (!P Q a b. happ (happ (happ (happ _1 P) Q) a) b <=> happ (happ P a) b /\ happ (happ Q a) b) ==> (!_0. (!P Q a. happ (happ (happ _0 P) Q) a = happ (happ (happ _1 P) Q) a) ==> (!l2 l1 Q P. 'h4/const/list/LIST_REL' (happ (happ _0 P) Q) l1 l2 <=> 'h4/const/list/LIST_REL' (happ _2 P) l1 l2 /\ 'h4/const/list/LIST_REL' (happ _2 Q) l1 l2))))
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_3f5864,V_3f5860]: ![F, G]: (![X]: s(V_3f5860,happ(s(fun(V_3f5864,V_3f5860),F),s(V_3f5864,X))) = s(V_3f5860,happ(s(fun(V_3f5864,V_3f5860),G),s(V_3f5864,X))) => s(fun(V_3f5864,V_3f5860),F) = s(fun(V_3f5864,V_3f5860),G))).
fof('h4/thm/list/LIST_REL_CONJ_', conjecture, ![A,B]: ![V__3]: (![P, A0, B0]: s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__3),s(fun(A,fun(B,bool)),P))),s(A,A0))),s(B,B0))) = s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,A0))),s(B,B0))) => ![V__2]: (![P, A0]: s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__2),s(fun(A,fun(B,bool)),P))),s(A,A0))) = s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__3),s(fun(A,fun(B,bool)),P))),s(A,A0))) => ![V__1]: (![P, Q, A0, B0]: (p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),happ(s(fun(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool)))),V__1),s(fun(A,fun(B,bool)),P))),s(fun(A,fun(B,bool)),Q))),s(A,A0))),s(B,B0)))) <=> (p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,A0))),s(B,B0)))) & p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),Q),s(A,A0))),s(B,B0)))))) => ![V__0]: (![P, Q, A0]: s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),happ(s(fun(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool)))),V__0),s(fun(A,fun(B,bool)),P))),s(fun(A,fun(B,bool)),Q))),s(A,A0))) = s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),happ(s(fun(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool)))),V__1),s(fun(A,fun(B,bool)),P))),s(fun(A,fun(B,bool)),Q))),s(A,A0))) => ![L2, L1, Q, P]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),happ(s(fun(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool)))),V__0),s(fun(A,fun(B,bool)),P))),s(fun(A,fun(B,bool)),Q))),s('h4/type/list/list'(A),L1),s('h4/type/list/list'(B),L2)))) <=> (p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__2),s(fun(A,fun(B,bool)),P))),s('h4/type/list/list'(A),L1),s('h4/type/list/list'(B),L2)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__2),s(fun(A,fun(B,bool)),Q))),s('h4/type/list/list'(A),L1),s('h4/type/list/list'(B),L2))))))))))).
