%   ORIGINAL: 'h4/thm/while/LEAST_ELIM_'
% 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) /\ (!n. (!m. 'h4/const/prim_rec/<' m n ==> ~P m) /\ P n ==> Q n) ==> Q ('h4/const/while/LEAST' 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) /\ (!n. (!m. 'h4/const/prim_rec/<' m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q ('h4/const/while/LEAST' 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_3f41509,V_3f41505]: ![F, G]: (![X]: s(V_3f41505,happ(s(fun(V_3f41509,V_3f41505),F),s(V_3f41509,X))) = s(V_3f41505,happ(s(fun(V_3f41509,V_3f41505),G),s(V_3f41509,X))) => s(fun(V_3f41509,V_3f41505),F) = s(fun(V_3f41509,V_3f41505),G))).
fof('h4/thm/while/LEAST_ELIM_', conjecture, ![Q, P]: ((?[N]: p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N)))) & ![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/num/num',bool),P),s('h4/type/num/num',N))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),Q),s('h4/type/num/num',N)))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),Q),s('h4/type/num/num','h4/const/while/LEAST'(s(fun('h4/type/num/num',bool),P)))))))).
