%   ORIGINAL: h4/words/word__le__n2w
% 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/TRUTH: T
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/words/dimword__def: h4/words/dimword h4/bool/the__value = h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/fcp/dimindex h4/bool/the__value)
% Assm: h4/words/w2n__n2w: !n. h4/words/w2n (h4/words/n2w n) = h4/arithmetic/MOD n (h4/words/dimword h4/bool/the__value)
% Assm: h4/words/word__msb__n2w: !n. h4/words/word__msb (h4/words/n2w n) <=> h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n
% Assm: h4/words/WORD__LE: !b a. h4/words/word__le a b <=> (h4/words/word__msb a <=> h4/words/word__msb b) /\ h4/arithmetic/_3C_3D (h4/words/w2n a) (h4/words/w2n b) \/ h4/words/word__msb a /\ ~h4/words/word__msb b
% Goal: !b a. h4/words/word__le (h4/words/n2w a) (h4/words/n2w b) <=> h4/bool/LET (h4/bool/LET (\sa sb. (sa <=> sb) /\ h4/arithmetic/_3C_3D (h4/arithmetic/MOD a (h4/words/dimword h4/bool/the__value)) (h4/arithmetic/MOD b (h4/words/dimword h4/bool/the__value)) \/ sa /\ ~sb) (h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) a)) (h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) b)
%   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_TRUTH]: T
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_wordss_dimwordu_u_def]: h4/words/dimword h4/bool/the__value = h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/fcp/dimindex h4/bool/the__value)
% Assm [h4s_wordss_w2nu_u_n2w]: !n. h4/words/w2n (h4/words/n2w n) = h4/arithmetic/MOD n (h4/words/dimword h4/bool/the__value)
% Assm [h4s_wordss_wordu_u_msbu_u_n2w]: !n. h4/words/word__msb (h4/words/n2w n) <=> h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n
% Assm [h4s_wordss_WORDu_u_LE]: !b a. h4/words/word__le a b <=> (h4/words/word__msb a <=> h4/words/word__msb b) /\ h4/arithmetic/_3C_3D (h4/words/w2n a) (h4/words/w2n b) \/ h4/words/word__msb a /\ ~h4/words/word__msb b
% Goal: !_1. (!a b sa sb. happ (happ (happ (happ _1 a) b) sa) sb <=> (sa <=> sb) /\ h4/arithmetic/_3C_3D (h4/arithmetic/MOD a (h4/words/dimword h4/bool/the__value)) (h4/arithmetic/MOD b (h4/words/dimword h4/bool/the__value)) \/ sa /\ ~sb) ==> (!_0. (!a b sa. happ (happ (happ _0 a) b) sa = happ (happ (happ _1 a) b) sa) ==> (!b a. h4/words/word__le (h4/words/n2w a) (h4/words/n2w b) <=> h4/bool/LET (h4/bool/LET (happ (happ _0 a) b) (h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) a)) (h4/bit/BIT (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) b)))
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_Q128825,TV_Q128821]: ![V_f, V_g]: (![V_x]: s(TV_Q128821,happ(s(t_fun(TV_Q128825,TV_Q128821),V_f),s(TV_Q128825,V_x))) = s(TV_Q128821,happ(s(t_fun(TV_Q128825,TV_Q128821),V_g),s(TV_Q128825,V_x))) => s(t_fun(TV_Q128825,TV_Q128821),V_f) = s(t_fun(TV_Q128825,TV_Q128821),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(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),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
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_wordss_dimwordu_u_def, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_wordss_dimword(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value)))))).
fof(ah4s_wordss_w2nu_u_n2w, axiom, ![TV_u_27a]: ![V_n]: s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_wordss_dimword(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value)))))).
fof(ah4s_wordss_wordu_u_msbu_u_n2w, axiom, ![TV_u_27a]: ![V_n]: s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_wordss_WORDu_u_LE, axiom, ![TV_u_27a]: ![V_b, V_a]: (p(s(t_bool,h4s_wordss_wordu_u_le(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b)))) <=> ((s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a))) = s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a))),s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b))))))) | (p(s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a)))) & ~ (p(s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b))))))))).
fof(ch4s_wordss_wordu_u_leu_u_n2w, conjecture, ![TV_u_27a]: ![V_uu_1]: (![V_a, V_b, V_sa, V_sb]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_b))),s(t_bool,V_sa))),s(t_bool,V_sb)))) <=> ((s(t_bool,V_sa) = s(t_bool,V_sb) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_wordss_dimword(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,h4s_wordss_dimword(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))))))) | (p(s(t_bool,V_sa)) & ~ (p(s(t_bool,V_sb)))))) => ![V_uu_0]: (![V_a, V_b, V_sa]: s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool)))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_b))),s(t_bool,V_sa))) = s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_b))),s(t_bool,V_sa))) => ![V_b, V_a]: s(t_bool,h4s_wordss_wordu_u_le(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_a))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_b))))) = s(t_bool,h4s_bools_let(s(t_fun(t_bool,t_bool),h4s_bools_let(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_bool,t_bool)))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_b))),s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_a))))),s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_b)))))))).
