%   ORIGINAL: 'h4/thm/while/OWHILE_IND_'
% 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: !P G f. (!s. ~G s ==> P s s) /\ (!s1 s2. G s1 /\ P (f s1) s2 ==> P s1 s2) ==> (!s1 s2. 'h4/const/while/OWHILE' G f s1 = 'h4/const/option/SOME' s2 ==> P s1 s2)
%   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: !P G f. (!s. ~happ G s ==> happ (happ P s) s) /\ (!s1 s2. happ G s1 /\ happ (happ P (happ f s1)) s2 ==> happ (happ P s1) s2) ==> (!s1 s2. 'h4/const/while/OWHILE' G f s1 = 'h4/const/option/SOME' s2 ==> happ (happ P s1) s2)
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_3f41909,V_3f41905]: ![F, G]: (![X]: s(V_3f41905,happ(s(fun(V_3f41909,V_3f41905),F),s(V_3f41909,X))) = s(V_3f41905,happ(s(fun(V_3f41909,V_3f41905),G),s(V_3f41909,X))) => s(fun(V_3f41909,V_3f41905),F) = s(fun(V_3f41909,V_3f41905),G))).
fof('h4/thm/while/OWHILE_IND_', conjecture, ![A]: ![P, G, F]: ((![S]: (~ (p(s(bool,happ(s(fun(A,bool),G),s(A,S))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,S))),s(A,S))))) & ![S1, S2]: ((p(s(bool,happ(s(fun(A,bool),G),s(A,S1)))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,happ(s(fun(A,A),F),s(A,S1))))),s(A,S2))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,S1))),s(A,S2)))))) => ![S1, S2]: (s('h4/type/option/option'(A),'h4/const/while/OWHILE'(s(fun(A,bool),G),s(fun(A,A),F),s(A,S1))) = s('h4/type/option/option'(A),'h4/const/option/SOME'(s(A,S2))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,S1))),s(A,S2))))))).
