%   ORIGINAL: 'h4/thm/list/FOLDL_UNION_BIGUNION_paired_'
% 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: !f ls s. 'h4/const/list/FOLDL' (\s0. 'h4/const/pair/UNCURRY' (\x y. 'h4/const/pred_set/UNION' s0 (f x y))) s ls = 'h4/const/pred_set/UNION' s ('h4/const/pred_set/BIGUNION' ('h4/const/pred_set/IMAGE' ('h4/const/pair/UNCURRY' f) ('h4/const/list/LIST_TO_SET' ls)))
%   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: !_2. (!s0 f x y. happ (happ (happ (happ _2 s0) f) x) y = 'h4/const/pred_set/UNION' s0 (happ (happ f x) y)) ==> (!_1. (!s0 f x. happ (happ (happ _1 s0) f) x = happ (happ (happ _2 s0) f) x) ==> (!_0. (!f s0. happ (happ _0 f) s0 = 'h4/const/pair/UNCURRY' (happ (happ _1 s0) f)) ==> (!f ls s. 'h4/const/list/FOLDL' (happ _0 f) s ls = 'h4/const/pred_set/UNION' s ('h4/const/pred_set/BIGUNION' ('h4/const/pred_set/IMAGE' ('h4/const/pair/UNCURRY' f) ('h4/const/list/LIST_TO_SET' ls))))))
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_3f9866,V_3f9862]: ![F, G]: (![X]: s(V_3f9862,happ(s(fun(V_3f9866,V_3f9862),F),s(V_3f9866,X))) = s(V_3f9862,happ(s(fun(V_3f9866,V_3f9862),G),s(V_3f9866,X))) => s(fun(V_3f9866,V_3f9862),F) = s(fun(V_3f9866,V_3f9862),G))).
fof('h4/thm/list/FOLDL_UNION_BIGUNION_paired_', conjecture, ![C,A,B]: ![V__2]: (![S0, F, X, Y]: s(fun(C,bool),happ(s(fun(B,fun(C,bool)),happ(s(fun(A,fun(B,fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool)))),happ(s(fun(fun(C,bool),fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool))))),V__2),s(fun(C,bool),S0))),s(fun(A,fun(B,fun(C,bool))),F))),s(A,X))),s(B,Y))) = s(fun(C,bool),'h4/const/pred_set/UNION'(s(fun(C,bool),S0),s(fun(C,bool),happ(s(fun(B,fun(C,bool)),happ(s(fun(A,fun(B,fun(C,bool))),F),s(A,X))),s(B,Y))))) => ![V__1]: (![S0, F, X]: s(fun(B,fun(C,bool)),happ(s(fun(A,fun(B,fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool)))),happ(s(fun(fun(C,bool),fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool))))),V__1),s(fun(C,bool),S0))),s(fun(A,fun(B,fun(C,bool))),F))),s(A,X))) = s(fun(B,fun(C,bool)),happ(s(fun(A,fun(B,fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool)))),happ(s(fun(fun(C,bool),fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool))))),V__2),s(fun(C,bool),S0))),s(fun(A,fun(B,fun(C,bool))),F))),s(A,X))) => ![V__0]: (![F, S0]: s(fun('h4/type/pair/prod'(A,B),fun(C,bool)),happ(s(fun(fun(C,bool),fun('h4/type/pair/prod'(A,B),fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(fun(C,bool),fun('h4/type/pair/prod'(A,B),fun(C,bool)))),V__0),s(fun(A,fun(B,fun(C,bool))),F))),s(fun(C,bool),S0))) = s(fun('h4/type/pair/prod'(A,B),fun(C,bool)),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool)))),happ(s(fun(fun(C,bool),fun(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,fun(C,bool))))),V__1),s(fun(C,bool),S0))),s(fun(A,fun(B,fun(C,bool))),F))))) => ![F, Ls, S]: s(fun(C,bool),'h4/const/list/FOLDL'(s(fun(fun(C,bool),fun('h4/type/pair/prod'(A,B),fun(C,bool))),happ(s(fun(fun(A,fun(B,fun(C,bool))),fun(fun(C,bool),fun('h4/type/pair/prod'(A,B),fun(C,bool)))),V__0),s(fun(A,fun(B,fun(C,bool))),F))),s(fun(C,bool),S),s('h4/type/list/list'('h4/type/pair/prod'(A,B)),Ls))) = s(fun(C,bool),'h4/const/pred_set/UNION'(s(fun(C,bool),S),s(fun(C,bool),'h4/const/pred_set/BIGUNION'(s(fun(fun(C,bool),bool),'h4/const/pred_set/IMAGE'(s(fun('h4/type/pair/prod'(A,B),fun(C,bool)),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,fun(C,bool))),F))),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'('h4/type/pair/prod'(A,B)),Ls))))))))))))).
