%   ORIGINAL: h4/basis__emit/i2w__itself__def
% 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/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/basis__emit/i2w__itself__primitive__def: h4/basis__emit/i2w__itself = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\i2w__itself a. h4/pair/pair__CASE a (\i v1. h4/combin/I (h4/integer__word/i2w i)))
% Goal: !i. h4/basis__emit/i2w__itself (h4/pair/_2C i h4/bool/the__value) = h4/integer__word/i2w i
%   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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_basisu_u_emits_i2wu_u_itselfu_u_primitiveu_u_def]: !_4. (!i v1. happ (happ _4 i) v1 = h4/combin/I (h4/integer__word/i2w i)) ==> (!_3. (!i. happ _3 i = happ _4 i) ==> (!_2. (!a. happ _2 a = h4/pair/pair__CASE a _3) ==> (!_1. (!i2w__itself. happ _1 i2w__itself = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/basis__emit/i2w__itself = h4/relation/WFREC (h4/min/_40 _0) _1))))
% Goal: !i. happ h4/basis__emit/i2w__itself (h4/pair/_2C i h4/bool/the__value) = h4/integer__word/i2w i
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_Q218323,TV_Q218319]: ![V_f, V_g]: (![V_x]: s(TV_Q218319,happ(s(t_fun(TV_Q218323,TV_Q218319),V_f),s(TV_Q218323,V_x))) = s(TV_Q218319,happ(s(t_fun(TV_Q218323,TV_Q218319),V_g),s(TV_Q218323,V_x))) => s(t_fun(TV_Q218323,TV_Q218319),V_f) = s(t_fun(TV_Q218323,TV_Q218319),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (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_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_basisu_u_emits_i2wu_u_itselfu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_4]: (![V_i, V_v1]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),happ(s(t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_4),s(t_h4s_integers_int,V_i))),s(t_h4s_bools_itself(TV_u_27a),V_v1))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_combins_i(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_integeru_u_words_i2w(s(t_h4s_integers_int,V_i))))) => ![V_uu_3]: (![V_i]: s(t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_3),s(t_h4s_integers_int,V_i))) = s(t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_4),s(t_h4s_integers_int,V_i))) => ![V_uu_2]: (![V_a]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),V_uu_2),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),V_a))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),V_a),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_bools_itself(TV_u_27a),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_3))) => ![V_uu_1]: (![V_i2wu_u_itself]: s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),V_i2wu_u_itself))) = s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_basisu_u_emits_i2wu_u_itself) = s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a))),V_uu_1))))))))).
fof(ch4s_basisu_u_emits_i2wu_u_itselfu_u_def, conjecture, ![TV_u_27a]: ![V_i]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_basisu_u_emits_i2wu_u_itself),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_bools_itself(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_integers_int,V_i),s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_integeru_u_words_i2w(s(t_h4s_integers_int,V_i)))).
