%   ORIGINAL: h4/words/FST__ADD__WITH__CARRY_c3
% 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/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/words/word__add__def: !w v. h4/words/word__add v w = h4/words/n2w (h4/arithmetic/_2B (h4/words/w2n v) (h4/words/w2n w))
% Assm: h4/words/add__with__carry__def: !y x carry__in. h4/words/add__with__carry (h4/pair/_2C x (h4/pair/_2C y carry__in)) = h4/bool/LET (\unsigned__sum. h4/bool/LET (\result. h4/bool/LET (h4/bool/LET (\carry__out overflow. h4/pair/_2C result (h4/pair/_2C carry__out overflow)) (~(h4/words/w2n result = unsigned__sum))) ((h4/words/word__msb x <=> h4/words/word__msb y) /\ ~(h4/words/word__msb x <=> h4/words/word__msb result))) (h4/words/n2w unsigned__sum)) (h4/arithmetic/_2B (h4/arithmetic/_2B (h4/words/w2n x) (h4/words/w2n y)) (h4/bool/COND carry__in (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0))
% Assm: h4/words/word__sub__def: !w v. h4/words/word__sub v w = h4/words/word__add v (h4/words/word__2comp w)
% Assm: h4/words/word__add__n2w: !n m. h4/words/word__add (h4/words/n2w m) (h4/words/n2w n) = h4/words/n2w (h4/arithmetic/_2B m n)
% Assm: h4/words/WORD__NOT__NOT: !a. h4/words/word__1comp (h4/words/word__1comp a) = a
% Assm: h4/words/WORD__ADD__0_c0: !w. h4/words/word__add w (h4/words/n2w h4/num/0) = w
% Assm: h4/words/WORD__ADD__0_c1: !w. h4/words/word__add (h4/words/n2w h4/num/0) w = w
% Assm: h4/words/WORD__ADD__ASSOC: !x w v. h4/words/word__add v (h4/words/word__add w x) = h4/words/word__add (h4/words/word__add v w) x
% Assm: h4/words/WORD__ADD__COMM: !w v. h4/words/word__add v w = h4/words/word__add w v
% Assm: h4/words/WORD__ADD__RINV: !w. h4/words/word__add w (h4/words/word__2comp w) = h4/words/n2w h4/num/0
% Assm: h4/words/WORD__NOT: !w. h4/words/word__1comp w = h4/words/word__sub (h4/words/word__2comp w) (h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Goal: !n a. h4/pair/FST (h4/words/add__with__carry (h4/pair/_2C a (h4/pair/_2C (h4/words/n2w n) T))) = h4/words/word__sub a (h4/words/word__1comp (h4/words/n2w n))
%   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_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_wordss_wordu_u_addu_u_def]: !w v. h4/words/word__add v w = h4/words/n2w (h4/arithmetic/_2B (h4/words/w2n v) (h4/words/w2n w))
% Assm [h4s_wordss_addu_u_withu_u_carryu_u_def]: !_3. (!result carry__out overflow. happ (happ (happ _3 result) carry__out) overflow = h4/pair/_2C result (h4/pair/_2C carry__out overflow)) ==> (!_2. (!result carry__out. happ (happ _2 result) carry__out = happ (happ _3 result) carry__out) ==> (!_1. (!unsigned__sum y x result. ?v. (v <=> (h4/words/word__msb x <=> h4/words/word__msb y) /\ ~(h4/words/word__msb x <=> h4/words/word__msb result)) /\ (?v'. (v' <=> ~(h4/words/w2n result = unsigned__sum)) /\ happ (happ (happ (happ _1 unsigned__sum) y) x) result = h4/bool/LET (h4/bool/LET (happ _2 result) v') v)) ==> (!_0. (!y x unsigned__sum. happ (happ (happ _0 y) x) unsigned__sum = h4/bool/LET (happ (happ (happ _1 unsigned__sum) y) x) (h4/words/n2w unsigned__sum)) ==> (!y x carry__in. h4/words/add__with__carry (h4/pair/_2C x (h4/pair/_2C y carry__in)) = h4/bool/LET (happ (happ _0 y) x) (h4/arithmetic/_2B (h4/arithmetic/_2B (h4/words/w2n x) (h4/words/w2n y)) (h4/bool/COND carry__in (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0))))))
% Assm [h4s_wordss_wordu_u_subu_u_def]: !w v. h4/words/word__sub v w = h4/words/word__add v (h4/words/word__2comp w)
% Assm [h4s_wordss_wordu_u_addu_u_n2w]: !n m. h4/words/word__add (h4/words/n2w m) (h4/words/n2w n) = h4/words/n2w (h4/arithmetic/_2B m n)
% Assm [h4s_wordss_WORDu_u_NOTu_u_NOT]: !a. h4/words/word__1comp (h4/words/word__1comp a) = a
% Assm [h4s_wordss_WORDu_u_ADDu_u_0u_c0]: !w. h4/words/word__add w (h4/words/n2w h4/num/0) = w
% Assm [h4s_wordss_WORDu_u_ADDu_u_0u_c1]: !w. h4/words/word__add (h4/words/n2w h4/num/0) w = w
% Assm [h4s_wordss_WORDu_u_ADDu_u_ASSOC]: !x w v. h4/words/word__add v (h4/words/word__add w x) = h4/words/word__add (h4/words/word__add v w) x
% Assm [h4s_wordss_WORDu_u_ADDu_u_COMM]: !w v. h4/words/word__add v w = h4/words/word__add w v
% Assm [h4s_wordss_WORDu_u_ADDu_u_RINV]: !w. h4/words/word__add w (h4/words/word__2comp w) = h4/words/n2w h4/num/0
% Assm [h4s_wordss_WORDu_u_NOT]: !w. h4/words/word__1comp w = h4/words/word__sub (h4/words/word__2comp w) (h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Goal: !n a. h4/pair/FST (h4/words/add__with__carry (h4/pair/_2C a (h4/pair/_2C (h4/words/n2w n) T))) = h4/words/word__sub a (h4/words/word__1comp (h4/words/n2w n))
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_Q128615,TV_Q128611]: ![V_f, V_g]: (![V_x]: s(TV_Q128611,happ(s(t_fun(TV_Q128615,TV_Q128611),V_f),s(TV_Q128615,V_x))) = s(TV_Q128611,happ(s(t_fun(TV_Q128615,TV_Q128611),V_g),s(TV_Q128615,V_x))) => s(t_fun(TV_Q128615,TV_Q128611),V_f) = s(t_fun(TV_Q128615,TV_Q128611),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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, 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_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_wordss_wordu_u_addu_u_def, axiom, ![TV_u_27a]: ![V_w, V_v]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v))),s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))))))).
fof(ah4s_wordss_addu_u_withu_u_carryu_u_def, axiom, ![TV_u_27a]: ![V_uu_3]: (![V_result, V_carryu_u_out, V_overflow]: s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),happ(s(t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_3),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))),s(t_bool,V_carryu_u_out))),s(t_bool,V_overflow))) = s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result),s(t_h4s_pairs_prod(t_bool,t_bool),h4s_pairs_u_2c(s(t_bool,V_carryu_u_out),s(t_bool,V_overflow))))) => ![V_uu_2]: (![V_result, V_carryu_u_out]: s(t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_2),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))),s(t_bool,V_carryu_u_out))) = s(t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_3),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))),s(t_bool,V_carryu_u_out))) => ![V_uu_1]: (![V_unsignedu_u_sum, V_y, V_x, V_result]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))) = s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))) & ~ (s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))) = s(t_bool,h4s_wordss_wordu_u_msb(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result)))))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> ~ (s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))) = s(t_h4s_nums_num,V_unsignedu_u_sum))) & s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))))),V_uu_1),s(t_h4s_nums_num,V_unsignedu_u_sum))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))) = s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),h4s_bools_let(s(t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),h4s_bools_let(s(t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_bool,t_fun(t_bool,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_2),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_result))),s(t_bool,V_vi_))),s(t_bool,V_v))))) => ![V_uu_0]: (![V_y, V_x, V_unsignedu_u_sum]: s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_0),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))),s(t_h4s_nums_num,V_unsignedu_u_sum))) = s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),h4s_bools_let(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))))),V_uu_1),s(t_h4s_nums_num,V_unsignedu_u_sum))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_unsignedu_u_sum))))) => ![V_y, V_x, V_carryu_u_in]: s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),h4s_wordss_addu_u_withu_u_carry(s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x),s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y),s(t_bool,V_carryu_u_in))))))) = s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool)))),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_pairs_prod(t_bool,t_bool))))),V_uu_0),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))),s(t_h4s_nums_num,h4s_wordss_w2n(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_y))))),s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_carryu_u_in),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,h4s_nums_0)))))))))))).
fof(ah4s_wordss_wordu_u_subu_u_def, axiom, ![TV_u_27a]: ![V_w, V_v]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_sub(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_2comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))))).
fof(ah4s_wordss_wordu_u_addu_u_n2w, axiom, ![TV_u_27a]: ![V_n, V_m]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_m))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_wordss_WORDu_u_NOTu_u_NOT, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_1comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_1comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a)).
fof(ah4s_wordss_WORDu_u_ADDu_u_0u_c0, axiom, ![TV_u_27a]: ![V_w]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)).
fof(ah4s_wordss_WORDu_u_ADDu_u_0u_c1, axiom, ![TV_u_27a]: ![V_w]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)).
fof(ah4s_wordss_WORDu_u_ADDu_u_ASSOC, axiom, ![TV_u_27a]: ![V_x, V_w, V_v]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_x)))).
fof(ah4s_wordss_WORDu_u_ADDu_u_COMM, axiom, ![TV_u_27a]: ![V_w, V_v]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_v)))).
fof(ah4s_wordss_WORDu_u_ADDu_u_RINV, axiom, ![TV_u_27a]: ![V_w]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_add(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_2comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_wordss_WORDu_u_NOT, axiom, ![TV_u_27a]: ![V_w]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_1comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_sub(s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_wordu_u_2comp(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ch4s_wordss_FSTu_u_ADDu_u_WITHu_u_CARRYu_c3, conjecture, ![TV_u_27b]: ![V_n, V_a]: s(t_h4s_fcps_cart(t_bool,TV_u_27b),h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27b),t_h4s_pairs_prod(t_bool,t_bool)),h4s_wordss_addu_u_withu_u_carry(s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27b),t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27b),t_bool)),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27b),V_a),s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27b),t_bool),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27b),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))),s(t_bool,t))))))))) = s(t_h4s_fcps_cart(t_bool,TV_u_27b),h4s_wordss_wordu_u_sub(s(t_h4s_fcps_cart(t_bool,TV_u_27b),V_a),s(t_h4s_fcps_cart(t_bool,TV_u_27b),h4s_wordss_wordu_u_1comp(s(t_h4s_fcps_cart(t_bool,TV_u_27b),h4s_wordss_n2w(s(t_h4s_nums_num,V_n)))))))).
