%   ORIGINAL: 'h4/thm/while/OLEAST_INTRO_'
% 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: !Q P. ((!n. ~P n) ==> Q 'h4/const/option/NONE') /\ (!n. P n /\ (!m. 'h4/const/prim_rec/<' m n ==> ~P m) ==> Q ('h4/const/option/SOME' n)) ==> Q ('h4/const/while/OLEAST' P)
%   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: !Q P. ((!n. ~happ P n) ==> happ Q 'h4/const/option/NONE') /\ (!n. happ P n /\ (!m. 'h4/const/prim_rec/<' m n ==> ~happ P m) ==> happ Q ('h4/const/option/SOME' n)) ==> happ Q ('h4/const/while/OLEAST' P)
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_3f41659,V_3f41655]: ![F, G]: (![X]: s(V_3f41655,happ(s(fun(V_3f41659,V_3f41655),F),s(V_3f41659,X))) = s(V_3f41655,happ(s(fun(V_3f41659,V_3f41655),G),s(V_3f41659,X))) => s(fun(V_3f41659,V_3f41655),F) = s(fun(V_3f41659,V_3f41655),G))).
fof('h4/thm/while/OLEAST_INTRO_', conjecture, ![Q, P]: (((![N]: ~ (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N))))) => p(s(bool,happ(s(fun('h4/type/option/option'('h4/type/num/num'),bool),Q),s('h4/type/option/option'('h4/type/num/num'),'h4/const/option/NONE'))))) & ![N]: ((p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N)))) & ![M]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',M),s('h4/type/num/num',N)))) => ~ (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',M))))))) => p(s(bool,happ(s(fun('h4/type/option/option'('h4/type/num/num'),bool),Q),s('h4/type/option/option'('h4/type/num/num'),'h4/const/option/SOME'(s('h4/type/num/num',N)))))))) => p(s(bool,happ(s(fun('h4/type/option/option'('h4/type/num/num'),bool),Q),s('h4/type/option/option'('h4/type/num/num'),'h4/const/while/OLEAST'(s(fun('h4/type/num/num',bool),P)))))))).
