%   ORIGINAL: 'h4/thm/list/LIST_REL_EL_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: !R l1 l2. 'h4/const/list/LIST_REL' R l1 l2 <=> 'h4/const/list/LENGTH' l1 = 'h4/const/list/LENGTH' l2 /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l1) ==> R ('h4/const/list/EL' n l1) ('h4/const/list/EL' n 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: !R l1 l2. 'h4/const/list/LIST_REL' R l1 l2 <=> 'h4/const/list/LENGTH' l1 = 'h4/const/list/LENGTH' l2 /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l1) ==> happ (happ R ('h4/const/list/EL' n l1)) ('h4/const/list/EL' n 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_3f5614,V_3f5610]: ![F, G]: (![X]: s(V_3f5610,happ(s(fun(V_3f5614,V_3f5610),F),s(V_3f5614,X))) = s(V_3f5610,happ(s(fun(V_3f5614,V_3f5610),G),s(V_3f5614,X))) => s(fun(V_3f5614,V_3f5610),F) = s(fun(V_3f5614,V_3f5610),G))).
fof('h4/thm/list/LIST_REL_EL_EQN_', conjecture, ![A,B]: ![R, L1, L2]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(B,bool)),R),s('h4/type/list/list'(A),L1),s('h4/type/list/list'(B),L2)))) <=> (s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L1))) = s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(B),L2))) & ![N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L1)))))) => p(s(bool,happ(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),R),s(A,'h4/const/list/EL'(s('h4/type/num/num',N),s('h4/type/list/list'(A),L1))))),s(B,'h4/const/list/EL'(s('h4/type/num/num',N),s('h4/type/list/list'(B),L2)))))))))).
