%   ORIGINAL: h4/finite__map/is__fmap__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__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/finite__map/is__fmap__def: h4/finite__map/is__fmap = (\a0. !is__fmap_27. (!a00. a00 = (\a. h4/sum/INR h4/one/one0) \/ (?f a b. a00 = (\x. h4/bool/COND (x = a) (h4/sum/INL b) (f x)) /\ is__fmap_27 f) ==> is__fmap_27 a00) ==> is__fmap_27 a0)
% Goal: !a0. h4/finite__map/is__fmap a0 <=> a0 = (\a. h4/sum/INR h4/one/one0) \/ (?f a b. a0 = (\x. h4/bool/COND (x = a) (h4/sum/INL b) (f x)) /\ h4/finite__map/is__fmap f)
%   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_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% 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_finiteu_u_maps_isu_u_fmapu_u_def]: !x. h4/finite__map/is__fmap x <=> (!is__fmap_27. (!a00. (!x. happ a00 x = h4/sum/INR h4/one/one0) \/ (?f a b. (!x. ?v. (v <=> x = a) /\ happ a00 x = h4/bool/COND v (h4/sum/INL b) (happ f x)) /\ happ is__fmap_27 f) ==> happ is__fmap_27 a00) ==> happ is__fmap_27 x)
% Goal: !a0. h4/finite__map/is__fmap a0 <=> (!x. happ a0 x = h4/sum/INR h4/one/one0) \/ (?f a b. (!x. ?v. (v <=> x = a) /\ happ a0 x = h4/bool/COND v (h4/sum/INL b) (happ f x)) /\ h4/finite__map/is__fmap f)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q72452,TV_Q72448]: ![V_f, V_g]: (![V_x]: s(TV_Q72448,happ(s(t_fun(TV_Q72452,TV_Q72448),V_f),s(TV_Q72452,V_x))) = s(TV_Q72448,happ(s(t_fun(TV_Q72452,TV_Q72448),V_g),s(TV_Q72452,V_x))) => s(t_fun(TV_Q72452,TV_Q72448),V_f) = s(t_fun(TV_Q72452,TV_Q72448),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_OR, 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_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_finiteu_u_maps_isu_u_fmapu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,h4s_finiteu_u_maps_isu_u_fmap(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x)))) <=> ![V_isu_u_fmapu_27]: (![V_a00]: ((![V_x0]: s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0))) | ?[V_f, V_a, V_b]: (![V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x0) = s(TV_u_27a,V_a)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f),s(TV_u_27a,V_x0)))))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f)))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x))))))).
fof(ch4s_finiteu_u_maps_isu_u_fmapu_u_cases, conjecture, ![TV_u_27a,TV_u_27b]: ![V_a0]: (p(s(t_bool,h4s_finiteu_u_maps_isu_u_fmap(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a0)))) <=> (![V_x]: s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a0),s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0))) | ?[V_f, V_a, V_b]: (![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a0),s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f),s(TV_u_27a,V_x)))))) & p(s(t_bool,h4s_finiteu_u_maps_isu_u_fmap(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f)))))))).
