%   ORIGINAL: h4/fmaptree/relrec__cases
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__IMP: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/fmaptree/relrec__def: h4/fmaptree/relrec = (\h a0 a1. !relrec_27. (!a00 a10. (?i fm rfm. a00 = h4/fmaptree/FTNode i fm /\ a10 = h i rfm fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> relrec_27 (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d))) ==> relrec_27 a00 a10) ==> relrec_27 a0 a1)
% Goal: !h a1 a0. h4/fmaptree/relrec h a0 a1 <=> (?i fm rfm. a0 = h4/fmaptree/FTNode i fm /\ a1 = h i rfm fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)))
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_IMP]: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_fmaptrees_relrecu_u_def]: !x x x. h4/fmaptree/relrec x x x <=> (!relrec_27. (!a00 a10. (?i fm rfm. a00 = h4/fmaptree/FTNode i fm /\ a10 = happ (happ (happ x i) rfm) fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> happ (happ relrec_27 (h4/finite__map/FAPPLY fm d)) (h4/finite__map/FAPPLY rfm d))) ==> happ (happ relrec_27 a00) a10) ==> happ (happ relrec_27 x) x)
% Goal: !h a1 a0. h4/fmaptree/relrec h a0 a1 <=> (?i fm rfm. a0 = h4/fmaptree/FTNode i fm /\ a1 = happ (happ (happ h i) rfm) fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)))
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_Q255457,TV_Q255453]: ![V_f, V_g]: (![V_x]: s(TV_Q255453,happ(s(t_fun(TV_Q255457,TV_Q255453),V_f),s(TV_Q255457,V_x))) = s(TV_Q255453,happ(s(t_fun(TV_Q255457,TV_Q255453),V_g),s(TV_Q255457,V_x))) => s(t_fun(TV_Q255457,TV_Q255453),V_f) = s(t_fun(TV_Q255457,TV_Q255453),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_IMP, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_y)) => p(s(t_bool,V_x))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) => p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_ALL, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_fmaptrees_relrecu_u_def, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_x0, V_x1]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_x),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27c,V_x1)))) <=> ![V_relrecu_27]: (![V_a00, V_a10]: (?[V_i, V_fm, V_rfm]: (s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a00) = s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(TV_u_27c,V_a10) = s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_x),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a00))),s(TV_u_27c,V_a10))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_x0))),s(TV_u_27c,V_x1))))))).
fof(ch4s_fmaptrees_relrecu_u_cases, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_h, V_a1, V_a0]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0),s(TV_u_27c,V_a1)))) <=> ?[V_i, V_fm, V_rfm]: (s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0) = s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(TV_u_27c,V_a1) = s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))))))).
