%   ORIGINAL: 'h4/thm/list/EL_ALL_DISTINCT_EL_EQ_'
% 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: !l. 'h4/const/list/ALL_DISTINCT' l <=> (!n1 n2. 'h4/const/prim_rec/<' n1 ('h4/const/list/LENGTH' l) /\ 'h4/const/prim_rec/<' n2 ('h4/const/list/LENGTH' l) ==> ('h4/const/list/EL' n1 l = 'h4/const/list/EL' n2 l <=> n1 = n2))
%   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: !l. 'h4/const/list/ALL_DISTINCT' l <=> (!n1 n2. 'h4/const/prim_rec/<' n1 ('h4/const/list/LENGTH' l) /\ 'h4/const/prim_rec/<' n2 ('h4/const/list/LENGTH' l) ==> ('h4/const/list/EL' n1 l = 'h4/const/list/EL' n2 l <=> n1 = n2))
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_3f7690,V_3f7686]: ![F, G]: (![X]: s(V_3f7686,happ(s(fun(V_3f7690,V_3f7686),F),s(V_3f7690,X))) = s(V_3f7686,happ(s(fun(V_3f7690,V_3f7686),G),s(V_3f7690,X))) => s(fun(V_3f7690,V_3f7686),F) = s(fun(V_3f7690,V_3f7686),G))).
fof('h4/thm/list/EL_ALL_DISTINCT_EL_EQ_', conjecture, ![A]: ![L]: (p(s(bool,'h4/const/list/ALL_DISTINCT'(s('h4/type/list/list'(A),L)))) <=> ![N1, N2]: ((p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N1),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L)))))) & p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N2),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L))))))) => (s(A,'h4/const/list/EL'(s('h4/type/num/num',N1),s('h4/type/list/list'(A),L))) = s(A,'h4/const/list/EL'(s('h4/type/num/num',N2),s('h4/type/list/list'(A),L))) <=> s('h4/type/num/num',N1) = s('h4/type/num/num',N2))))).
