%   ORIGINAL: 'h4/thm/rich_list/MAP_SND_FILTER_NEQ_'
% 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: !z ls. 'h4/const/list/MAP' 'h4/const/pair/SND' ('h4/const/list/FILTER' ('h4/const/pair/UNCURRY' (\x y. ~(y = z))) ls) = 'h4/const/list/FILTER' (\y. ~(z = y)) ('h4/const/list/MAP' 'h4/const/pair/SND' 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. (!z y. happ (happ _2 z) y <=> ~(z = y)) ==> (!_1. (!z y. happ (happ _1 z) y <=> ~(y = z)) ==> (!_0. (!z x. happ (happ _0 z) x = happ _1 z) ==> (!z ls. 'h4/const/list/MAP' 'h4/const/pair/SND' ('h4/const/list/FILTER' ('h4/const/pair/UNCURRY' (happ _0 z)) ls) = 'h4/const/list/FILTER' (happ _2 z) ('h4/const/list/MAP' 'h4/const/pair/SND' 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_3f57811,V_3f57807]: ![F, G]: (![X]: s(V_3f57807,happ(s(fun(V_3f57811,V_3f57807),F),s(V_3f57811,X))) = s(V_3f57807,happ(s(fun(V_3f57811,V_3f57807),G),s(V_3f57811,X))) => s(fun(V_3f57811,V_3f57807),F) = s(fun(V_3f57811,V_3f57807),G))).
fof('h4/thm/rich_list/MAP_SND_FILTER_NEQ_', conjecture, ![B,A]: ![V__2]: (![Z, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),V__2),s(A,Z))),s(A,Y)))) <=> ~ (s(A,Z) = s(A,Y))) => ![V__1]: (![Z, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),V__1),s(A,Z))),s(A,Y)))) <=> ~ (s(A,Y) = s(A,Z))) => ![V__0]: (![Z, X]: s(fun(A,bool),happ(s(fun(B,fun(A,bool)),happ(s(fun(A,fun(B,fun(A,bool))),V__0),s(A,Z))),s(B,X))) = s(fun(A,bool),happ(s(fun(A,fun(A,bool)),V__1),s(A,Z))) => ![Z, Ls]: s('h4/type/list/list'(A),'h4/const/list/MAP'(s(fun('h4/type/pair/prod'(B,A),A),'h4/const/pair/SND'),s('h4/type/list/list'('h4/type/pair/prod'(B,A)),'h4/const/list/FILTER'(s(fun('h4/type/pair/prod'(B,A),bool),'h4/const/pair/UNCURRY'(s(fun(B,fun(A,bool)),happ(s(fun(A,fun(B,fun(A,bool))),V__0),s(A,Z))))),s('h4/type/list/list'('h4/type/pair/prod'(B,A)),Ls))))) = s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),V__2),s(A,Z))),s('h4/type/list/list'(A),'h4/const/list/MAP'(s(fun('h4/type/pair/prod'(B,A),A),'h4/const/pair/SND'),s('h4/type/list/list'('h4/type/pair/prod'(B,A)),Ls))))))))).
