%   ORIGINAL: 'h4/thm/option/IF_NONE_EQUALS_OPTION_e3'
% 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: !x X P. 'h4/const/bool/COND' P 'h4/const/option/NONE' X = 'h4/const/option/SOME' x <=> ~P /\ X = 'h4/const/option/SOME' x
%   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: !x X P. 'h4/const/bool/COND' P 'h4/const/option/NONE' X = 'h4/const/option/SOME' x <=> ~P /\ X = 'h4/const/option/SOME' x
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_3f100107,V_3f100103]: ![F, G]: (![X]: s(V_3f100103,happ(s(fun(V_3f100107,V_3f100103),F),s(V_3f100107,X))) = s(V_3f100103,happ(s(fun(V_3f100107,V_3f100103),G),s(V_3f100107,X))) => s(fun(V_3f100107,V_3f100103),F) = s(fun(V_3f100107,V_3f100103),G))).
fof('h4/thm/option/IF_NONE_EQUALS_OPTION_e3', conjecture, ![A]: ![X, X0, P]: (s('h4/type/option/option'(A),'h4/const/bool/COND'(s(bool,P),s('h4/type/option/option'(A),'h4/const/option/NONE'),s('h4/type/option/option'(A),X0))) = s('h4/type/option/option'(A),'h4/const/option/SOME'(s(A,X))) <=> (~ (p(s(bool,P))) & s('h4/type/option/option'(A),X0) = s('h4/type/option/option'(A),'h4/const/option/SOME'(s(A,X)))))).
