%   ORIGINAL: 'h4/thm/while/WHILE_RULE_'
% 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 R B C. 'h4/const/relation/WF' R /\ (!s. B s ==> R (C s) s) ==> 'h4/const/while/HOARE_SPEC' (\s. P s /\ B s) C P ==> 'h4/const/while/HOARE_SPEC' P ('h4/const/while/WHILE' B C) (\s. P s /\ ~B s)
%   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: !_1. (!B s. happ (happ _1 B) s <=> ~happ B s) ==> (!_0. (!P B s. happ (happ (happ _0 P) B) s <=> happ P s /\ happ B s) ==> (!P R B C. 'h4/const/relation/WF' R /\ (!s. happ B s ==> happ (happ R (happ C s)) s) ==> 'h4/const/while/HOARE_SPEC' (happ (happ _0 P) B) C P ==> 'h4/const/while/HOARE_SPEC' P ('h4/const/while/WHILE' B C) (happ (happ _0 P) (happ _1 B))))
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_3f41409,V_3f41405]: ![F, G]: (![X]: s(V_3f41405,happ(s(fun(V_3f41409,V_3f41405),F),s(V_3f41409,X))) = s(V_3f41405,happ(s(fun(V_3f41409,V_3f41405),G),s(V_3f41409,X))) => s(fun(V_3f41409,V_3f41405),F) = s(fun(V_3f41409,V_3f41405),G))).
fof('h4/thm/while/WHILE_RULE_', conjecture, ![A]: ![V__1]: (![B, S]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),B))),s(A,S)))) <=> ~ (p(s(bool,happ(s(fun(A,bool),B),s(A,S)))))) => ![V__0]: (![P, B, S]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,bool))),V__0),s(fun(A,bool),P))),s(fun(A,bool),B))),s(A,S)))) <=> (p(s(bool,happ(s(fun(A,bool),P),s(A,S)))) & p(s(bool,happ(s(fun(A,bool),B),s(A,S)))))) => ![P, R, B, C]: ((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(bool,'h4/const/while/HOARE_SPEC'(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,bool))),V__0),s(fun(A,bool),P))),s(fun(A,bool),B))),s(fun(A,A),C),s(fun(A,bool),P)))) => p(s(bool,'h4/const/while/HOARE_SPEC'(s(fun(A,bool),P),s(fun(A,A),'h4/const/while/WHILE'(s(fun(A,bool),B),s(fun(A,A),C))),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,bool))),V__0),s(fun(A,bool),P))),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),B))))))))))))).
