%   ORIGINAL: h4/hreal/HRAT__LT__TRANS
% 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
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/hrat/HRAT__ADD__ASSOC: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm: h4/hreal/hrat__lt0: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Goal: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_hrats_HRATu_u_ADDu_u_ASSOC]: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm [h4s_hreals_hratu_u_lt0]: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Goal: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q253631,TV_Q253627]: ![V_f, V_g]: (![V_x]: s(TV_Q253627,happ(s(t_fun(TV_Q253631,TV_Q253627),V_f),s(TV_Q253631,V_x))) = s(TV_Q253627,happ(s(t_fun(TV_Q253631,TV_Q253627),V_g),s(TV_Q253631,V_x))) => s(t_fun(TV_Q253631,TV_Q253627),V_f) = s(t_fun(TV_Q253631,TV_Q253627),V_g))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_hrats_HRATu_u_ADDu_u_ASSOC, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,V_j)))).
fof(ah4s_hreals_hratu_u_lt0, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) <=> ?[V_d]: s(t_h4s_hrats_hrat,V_y) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_d))))).
fof(ch4s_hreals_HRATu_u_LTu_u_TRANS, conjecture, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))))).
