%   ORIGINAL: 'h4/thm/while/WHILE_INDUCTION_'
% 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: !B C R. 'h4/const/relation/WF' R /\ (!s. B s ==> R (C s) s) ==> (!P. (!s. (B s ==> P (C s)) ==> P s) ==> (!v. P v))
%   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: !B C R. 'h4/const/relation/WF' R /\ (!s. happ B s ==> happ (happ R (happ C s)) s) ==> (!P. (!s. (happ B s ==> happ P (happ C s)) ==> happ P s) ==> (!v. happ P v))
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_3f41384,V_3f41380]: ![F, G]: (![X]: s(V_3f41380,happ(s(fun(V_3f41384,V_3f41380),F),s(V_3f41384,X))) = s(V_3f41380,happ(s(fun(V_3f41384,V_3f41380),G),s(V_3f41384,X))) => s(fun(V_3f41384,V_3f41380),F) = s(fun(V_3f41384,V_3f41380),G))).
fof('h4/thm/while/WHILE_INDUCTION_', conjecture, ![A]: ![B, C, R]: ((p(s(bool,'h4/const/relation/WF'(s(fun(A,fun(A,bool)),R)))) & ![S]: (p(s(bool,happ(s(fun(A,bool),B),s(A,S)))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun(A,A),C),s(A,S))))),s(A,S)))))) => ![P]: (![S]: ((p(s(bool,happ(s(fun(A,bool),B),s(A,S)))) => p(s(bool,happ(s(fun(A,bool),P),s(A,happ(s(fun(A,A),C),s(A,S))))))) => p(s(bool,happ(s(fun(A,bool),P),s(A,S))))) => ![V]: p(s(bool,happ(s(fun(A,bool),P),s(A,V))))))).
