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forall p a b. a = b -> (exists pfa_offset_left_equal pfa_offset_right_equal. (a) + (p) * pfa_offset_left_equal = (b) + (p) * pfa_offset_right_equal)

forall p. (~((p) = 1) /\ forall pfa_factor_left_zero_domain pfa_factor_right_zero_domain. (p) = pfa_factor_left_zero_domain * pfa_factor_right_zero_domain -> pfa_factor_left_zero_domain = 1 \/ pfa_factor_right_zero_domain = 1) -> (exists pfa_gap_zero_bound. pfa_gap_zero_bound + S (0) = (p))

forall p a. (exists pfa_gap_reflexive_bound. pfa_gap_reflexive_bound + S (a) = (p)) -> (((exists pfa_gap_reflexivebound. pfa_gap_reflexivebound + S (a) = (p)) /\ ((exists pfa_offset_left_reflexivecongruence pfa_offset_right_reflexivecongruence. (a) + (p) * pfa_offset_left_reflexivecongruence = (a) + (p) * pfa_offset_right_reflexivecongruence))))

forall p n m r. n = m -> (((exists pfa_gap_input_equal_sourcebound. pfa_gap_input_equal_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_sourcecongruence pfa_offset_right_input_equal_sourcecongruence. (m) + (p) * pfa_offset_left_input_equal_sourcecongruence = (r) + (p) * pfa_offset_right_input_equal_sourcecongruence)))) -> (((exists pfa_gap_input_equal_targetbound. pfa_gap_input_equal_targetbound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_targetcongruence pfa_offset_right_input_equal_targetcongruence. (n) + (p) * pfa_offset_left_input_equal_targetcongruence = (r) + (p) * pfa_offset_right_input_equal_targetcongruence))))

forall p n m r. (exists pfa_offset_left_transport pfa_offset_right_transport. (n) + (p) * pfa_offset_left_transport = (m) + (p) * pfa_offset_right_transport) -> (((exists pfa_gap_transport_sourcebound. pfa_gap_transport_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_sourcecongruence pfa_offset_right_transport_sourcecongruence. (m) + (p) * pfa_offset_left_transport_sourcecongruence = (r) + (p) * pfa_offset_right_transport_sourcecongruence)))) -> (((exists pfa_gap_transport_resultbound. pfa_gap_transport_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_resultcongruence pfa_offset_right_transport_resultcongruence. (n) + (p) * pfa_offset_left_transport_resultcongruence = (r) + (p) * pfa_offset_right_transport_resultcongruence))))

forall p a r. (exists pfa_gap_bounded_value. pfa_gap_bounded_value + S (a) = (p)) -> (((exists pfa_gap_bounded_resultbound. pfa_gap_bounded_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_bounded_resultcongruence pfa_offset_right_bounded_resultcongruence. (a) + (p) * pfa_offset_left_bounded_resultcongruence = (r) + (p) * pfa_offset_right_bounded_resultcongruence)))) -> r = a

forall p. (~((p) = 1) /\ forall pfa_factor_left_modulus_domain pfa_factor_right_modulus_domain. (p) = pfa_factor_left_modulus_domain * pfa_factor_right_modulus_domain -> pfa_factor_left_modulus_domain = 1 \/ pfa_factor_right_modulus_domain = 1) -> (((exists pfa_gap_modulus_zerobound. pfa_gap_modulus_zerobound + S (0) = (p)) /\ ((exists pfa_offset_left_modulus_zerocongruence pfa_offset_right_modulus_zerocongruence. (p) + (p) * pfa_offset_left_modulus_zerocongruence = (0) + (p) * pfa_offset_right_modulus_zerocongruence))))

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_adddomain pfa_factor_right_adddomain. (p) = pfa_factor_left_adddomain * pfa_factor_right_adddomain -> pfa_factor_left_adddomain = 1 \/ pfa_factor_right_adddomain = 1) -> (exists pfa_gap_addleft. pfa_gap_addleft + S (a) = (p)) -> (exists pfa_gap_addright. pfa_gap_addright + S (b) = (p)) -> exists c. (((exists pfa_gap_addexistsleft. pfa_gap_addexistsleft + S (a) = (p)) /\ (((exists pfa_gap_addexistsright. pfa_gap_addexistsright + S (b) = (p)) /\ ((((exists pfa_gap_addexistsresultbound. pfa_gap_addexistsresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addexistsresultcongruence pfa_offset_right_addexistsresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addexistsresultcongruence = (c) + (p) * pfa_offset_right_addexistsresultcongruence)))))))))

forall p a b c d. (((exists pfa_gap_addfirstleft. pfa_gap_addfirstleft + S (a) = (p)) /\ (((exists pfa_gap_addfirstright. pfa_gap_addfirstright + S (b) = (p)) /\ ((((exists pfa_gap_addfirstresultbound. pfa_gap_addfirstresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addfirstresultcongruence pfa_offset_right_addfirstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addfirstresultcongruence = (c) + (p) * pfa_offset_right_addfirstresultcongruence))))))))) -> (((exists pfa_gap_addsecondleft. pfa_gap_addsecondleft + S (a) = (p)) /\ (((exists pfa_gap_addsecondright. pfa_gap_addsecondright + S (b) = (p)) /\ ((((exists pfa_gap_addsecondresultbound. pfa_gap_addsecondresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_addsecondresultcongruence pfa_offset_right_addsecondresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addsecondresultcongruence = (d) + (p) * pfa_offset_right_addsecondresultcongruence))))))))) -> c = d

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_addunique_domain pfa_factor_right_addunique_domain. (p) = pfa_factor_left_addunique_domain * pfa_factor_right_addunique_domain -> pfa_factor_left_addunique_domain = 1 \/ pfa_factor_right_addunique_domain = 1) -> (exists pfa_gap_addunique_left. pfa_gap_addunique_left + S (a) = (p)) -> (exists pfa_gap_addunique_right. pfa_gap_addunique_right + S (b) = (p)) -> exists c. (((exists pfa_gap_addchosenleft. pfa_gap_addchosenleft + S (a) = (p)) /\ (((exists pfa_gap_addchosenright. pfa_gap_addchosenright + S (b) = (p)) /\ ((((exists pfa_gap_addchosenresultbound. pfa_gap_addchosenresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addchosenresultcongruence pfa_offset_right_addchosenresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addchosenresultcongruence = (c) + (p) * pfa_offset_right_addchosenresultcongruence))))))))) /\ forall d. (((exists pfa_gap_addcomparisonleft. pfa_gap_addcomparisonleft + S (a) = (p)) /\ (((exists pfa_gap_addcomparisonright. pfa_gap_addcomparisonright + S (b) = (p)) /\ ((((exists pfa_gap_addcomparisonresultbound. pfa_gap_addcomparisonresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_addcomparisonresultcongruence pfa_offset_right_addcomparisonresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomparisonresultcongruence = (d) + (p) * pfa_offset_right_addcomparisonresultcongruence))))))))) -> d = c

forall p a b c. (((exists pfa_gap_addcomm_sourceleft. pfa_gap_addcomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_addcomm_sourceright. pfa_gap_addcomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_addcomm_sourceresultbound. pfa_gap_addcomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_sourceresultcongruence pfa_offset_right_addcomm_sourceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_addcomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_addcomm_targetleft. pfa_gap_addcomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_addcomm_targetright. pfa_gap_addcomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_addcomm_targetresultbound. pfa_gap_addcomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_targetresultcongruence pfa_offset_right_addcomm_targetresultcongruence. ((b) + (a)) + (p) * pfa_offset_left_addcomm_targetresultcongruence = (c) + (p) * pfa_offset_right_addcomm_targetresultcongruence)))))))))

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_multiplydomain pfa_factor_right_multiplydomain. (p) = pfa_factor_left_multiplydomain * pfa_factor_right_multiplydomain -> pfa_factor_left_multiplydomain = 1 \/ pfa_factor_right_multiplydomain = 1) -> (exists pfa_gap_multiplyleft. pfa_gap_multiplyleft + S (a) = (p)) -> (exists pfa_gap_multiplyright. pfa_gap_multiplyright + S (b) = (p)) -> exists c. (((exists pfa_gap_multiplyexistsleft. pfa_gap_multiplyexistsleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyexistsright. pfa_gap_multiplyexistsright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyexistsresultbound. pfa_gap_multiplyexistsresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyexistsresultcongruence pfa_offset_right_multiplyexistsresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyexistsresultcongruence = (c) + (p) * pfa_offset_right_multiplyexistsresultcongruence)))))))))

forall p a b c d. (((exists pfa_gap_multiplyfirstleft. pfa_gap_multiplyfirstleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyfirstright. pfa_gap_multiplyfirstright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyfirstresultbound. pfa_gap_multiplyfirstresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyfirstresultcongruence pfa_offset_right_multiplyfirstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyfirstresultcongruence = (c) + (p) * pfa_offset_right_multiplyfirstresultcongruence))))))))) -> (((exists pfa_gap_multiplysecondleft. pfa_gap_multiplysecondleft + S (a) = (p)) /\ (((exists pfa_gap_multiplysecondright. pfa_gap_multiplysecondright + S (b) = (p)) /\ ((((exists pfa_gap_multiplysecondresultbound. pfa_gap_multiplysecondresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_multiplysecondresultcongruence pfa_offset_right_multiplysecondresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplysecondresultcongruence = (d) + (p) * pfa_offset_right_multiplysecondresultcongruence))))))))) -> c = d

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_multiplyunique_domain pfa_factor_right_multiplyunique_domain. (p) = pfa_factor_left_multiplyunique_domain * pfa_factor_right_multiplyunique_domain -> pfa_factor_left_multiplyunique_domain = 1 \/ pfa_factor_right_multiplyunique_domain = 1) -> (exists pfa_gap_multiplyunique_left. pfa_gap_multiplyunique_left + S (a) = (p)) -> (exists pfa_gap_multiplyunique_right. pfa_gap_multiplyunique_right + S (b) = (p)) -> exists c. (((exists pfa_gap_multiplychosenleft. pfa_gap_multiplychosenleft + S (a) = (p)) /\ (((exists pfa_gap_multiplychosenright. pfa_gap_multiplychosenright + S (b) = (p)) /\ ((((exists pfa_gap_multiplychosenresultbound. pfa_gap_multiplychosenresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplychosenresultcongruence pfa_offset_right_multiplychosenresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplychosenresultcongruence = (c) + (p) * pfa_offset_right_multiplychosenresultcongruence))))))))) /\ forall d. (((exists pfa_gap_multiplycomparisonleft. pfa_gap_multiplycomparisonleft + S (a) = (p)) /\ (((exists pfa_gap_multiplycomparisonright. pfa_gap_multiplycomparisonright + S (b) = (p)) /\ ((((exists pfa_gap_multiplycomparisonresultbound. pfa_gap_multiplycomparisonresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_multiplycomparisonresultcongruence pfa_offset_right_multiplycomparisonresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplycomparisonresultcongruence = (d) + (p) * pfa_offset_right_multiplycomparisonresultcongruence))))))))) -> d = c

forall p a b c. (((exists pfa_gap_multiplycomm_sourceleft. pfa_gap_multiplycomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_multiplycomm_sourceright. pfa_gap_multiplycomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_multiplycomm_sourceresultbound. pfa_gap_multiplycomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplycomm_sourceresultcongruence pfa_offset_right_multiplycomm_sourceresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplycomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_multiplycomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_multiplycomm_targetleft. pfa_gap_multiplycomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_multiplycomm_targetright. pfa_gap_multiplycomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_multiplycomm_targetresultbound. pfa_gap_multiplycomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplycomm_targetresultcongruence pfa_offset_right_multiplycomm_targetresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_multiplycomm_targetresultcongruence = (c) + (p) * pfa_offset_right_multiplycomm_targetresultcongruence)))))))))

forall p a b c x y u v. (((exists pfa_gap_addassoc_firstleft. pfa_gap_addassoc_firstleft + S (a) = (p)) /\ (((exists pfa_gap_addassoc_firstright. pfa_gap_addassoc_firstright + S (b) = (p)) /\ ((((exists pfa_gap_addassoc_firstresultbound. pfa_gap_addassoc_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_addassoc_firstresultcongruence pfa_offset_right_addassoc_firstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addassoc_firstresultcongruence = (x) + (p) * pfa_offset_right_addassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_addassoc_leftleft. pfa_gap_addassoc_leftleft + S (x) = (p)) /\ (((exists pfa_gap_addassoc_leftright. pfa_gap_addassoc_leftright + S (c) = (p)) /\ ((((exists pfa_gap_addassoc_leftresultbound. pfa_gap_addassoc_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_addassoc_leftresultcongruence pfa_offset_right_addassoc_leftresultcongruence. ((x) + (c)) + (p) * pfa_offset_left_addassoc_leftresultcongruence = (u) + (p) * pfa_offset_right_addassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_addassoc_secondleft. pfa_gap_addassoc_secondleft + S (b) = (p)) /\ (((exists pfa_gap_addassoc_secondright. pfa_gap_addassoc_secondright + S (c) = (p)) /\ ((((exists pfa_gap_addassoc_secondresultbound. pfa_gap_addassoc_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_addassoc_secondresultcongruence pfa_offset_right_addassoc_secondresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_addassoc_secondresultcongruence = (y) + (p) * pfa_offset_right_addassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_addassoc_rightleft. pfa_gap_addassoc_rightleft + S (a) = (p)) /\ (((exists pfa_gap_addassoc_rightright. pfa_gap_addassoc_rightright + S (y) = (p)) /\ ((((exists pfa_gap_addassoc_rightresultbound. pfa_gap_addassoc_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addassoc_rightresultcongruence pfa_offset_right_addassoc_rightresultcongruence. ((a) + (y)) + (p) * pfa_offset_left_addassoc_rightresultcongruence = (v) + (p) * pfa_offset_right_addassoc_rightresultcongruence))))))))) -> u = v

forall p a b c x y u v. (((exists pfa_gap_multiplyassoc_firstleft. pfa_gap_multiplyassoc_firstleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyassoc_firstright. pfa_gap_multiplyassoc_firstright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyassoc_firstresultbound. pfa_gap_multiplyassoc_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_firstresultcongruence pfa_offset_right_multiplyassoc_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyassoc_firstresultcongruence = (x) + (p) * pfa_offset_right_multiplyassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_leftleft. pfa_gap_multiplyassoc_leftleft + S (x) = (p)) /\ (((exists pfa_gap_multiplyassoc_leftright. pfa_gap_multiplyassoc_leftright + S (c) = (p)) /\ ((((exists pfa_gap_multiplyassoc_leftresultbound. pfa_gap_multiplyassoc_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_leftresultcongruence pfa_offset_right_multiplyassoc_leftresultcongruence. ((x) * (c)) + (p) * pfa_offset_left_multiplyassoc_leftresultcongruence = (u) + (p) * pfa_offset_right_multiplyassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_secondleft. pfa_gap_multiplyassoc_secondleft + S (b) = (p)) /\ (((exists pfa_gap_multiplyassoc_secondright. pfa_gap_multiplyassoc_secondright + S (c) = (p)) /\ ((((exists pfa_gap_multiplyassoc_secondresultbound. pfa_gap_multiplyassoc_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_secondresultcongruence pfa_offset_right_multiplyassoc_secondresultcongruence. ((b) * (c)) + (p) * pfa_offset_left_multiplyassoc_secondresultcongruence = (y) + (p) * pfa_offset_right_multiplyassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_rightleft. pfa_gap_multiplyassoc_rightleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyassoc_rightright. pfa_gap_multiplyassoc_rightright + S (y) = (p)) /\ ((((exists pfa_gap_multiplyassoc_rightresultbound. pfa_gap_multiplyassoc_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_rightresultcongruence pfa_offset_right_multiplyassoc_rightresultcongruence. ((a) * (y)) + (p) * pfa_offset_left_multiplyassoc_rightresultcongruence = (v) + (p) * pfa_offset_right_multiplyassoc_rightresultcongruence))))))))) -> u = v

forall p a b c s x y u v. (((exists pfa_gap_leftdistribution_sumleft. pfa_gap_leftdistribution_sumleft + S (b) = (p)) /\ (((exists pfa_gap_leftdistribution_sumright. pfa_gap_leftdistribution_sumright + S (c) = (p)) /\ ((((exists pfa_gap_leftdistribution_sumresultbound. pfa_gap_leftdistribution_sumresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_leftdistribution_sumresultcongruence pfa_offset_right_leftdistribution_sumresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_leftdistribution_sumresultcongruence = (s) + (p) * pfa_offset_right_leftdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_leftleft. pfa_gap_leftdistribution_leftleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_leftright. pfa_gap_leftdistribution_leftright + S (s) = (p)) /\ ((((exists pfa_gap_leftdistribution_leftresultbound. pfa_gap_leftdistribution_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_leftdistribution_leftresultcongruence pfa_offset_right_leftdistribution_leftresultcongruence. ((a) * (s)) + (p) * pfa_offset_left_leftdistribution_leftresultcongruence = (u) + (p) * pfa_offset_right_leftdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_firstleft. pfa_gap_leftdistribution_firstleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_firstright. pfa_gap_leftdistribution_firstright + S (b) = (p)) /\ ((((exists pfa_gap_leftdistribution_firstresultbound. pfa_gap_leftdistribution_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_leftdistribution_firstresultcongruence pfa_offset_right_leftdistribution_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_leftdistribution_firstresultcongruence = (x) + (p) * pfa_offset_right_leftdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_secondleft. pfa_gap_leftdistribution_secondleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_secondright. pfa_gap_leftdistribution_secondright + S (c) = (p)) /\ ((((exists pfa_gap_leftdistribution_secondresultbound. pfa_gap_leftdistribution_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_leftdistribution_secondresultcongruence pfa_offset_right_leftdistribution_secondresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_leftdistribution_secondresultcongruence = (y) + (p) * pfa_offset_right_leftdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_rightleft. pfa_gap_leftdistribution_rightleft + S (x) = (p)) /\ (((exists pfa_gap_leftdistribution_rightright. pfa_gap_leftdistribution_rightright + S (y) = (p)) /\ ((((exists pfa_gap_leftdistribution_rightresultbound. pfa_gap_leftdistribution_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_leftdistribution_rightresultcongruence pfa_offset_right_leftdistribution_rightresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_leftdistribution_rightresultcongruence = (v) + (p) * pfa_offset_right_leftdistribution_rightresultcongruence))))))))) -> u = v

forall p a b c s x y u v. (((exists pfa_gap_rightdistribution_sumleft. pfa_gap_rightdistribution_sumleft + S (b) = (p)) /\ (((exists pfa_gap_rightdistribution_sumright. pfa_gap_rightdistribution_sumright + S (c) = (p)) /\ ((((exists pfa_gap_rightdistribution_sumresultbound. pfa_gap_rightdistribution_sumresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_rightdistribution_sumresultcongruence pfa_offset_right_rightdistribution_sumresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_rightdistribution_sumresultcongruence = (s) + (p) * pfa_offset_right_rightdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_leftleft. pfa_gap_rightdistribution_leftleft + S (s) = (p)) /\ (((exists pfa_gap_rightdistribution_leftright. pfa_gap_rightdistribution_leftright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_leftresultbound. pfa_gap_rightdistribution_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_rightdistribution_leftresultcongruence pfa_offset_right_rightdistribution_leftresultcongruence. ((s) * (a)) + (p) * pfa_offset_left_rightdistribution_leftresultcongruence = (u) + (p) * pfa_offset_right_rightdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_firstleft. pfa_gap_rightdistribution_firstleft + S (b) = (p)) /\ (((exists pfa_gap_rightdistribution_firstright. pfa_gap_rightdistribution_firstright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_firstresultbound. pfa_gap_rightdistribution_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_rightdistribution_firstresultcongruence pfa_offset_right_rightdistribution_firstresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_rightdistribution_firstresultcongruence = (x) + (p) * pfa_offset_right_rightdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_secondleft. pfa_gap_rightdistribution_secondleft + S (c) = (p)) /\ (((exists pfa_gap_rightdistribution_secondright. pfa_gap_rightdistribution_secondright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_secondresultbound. pfa_gap_rightdistribution_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_rightdistribution_secondresultcongruence pfa_offset_right_rightdistribution_secondresultcongruence. ((c) * (a)) + (p) * pfa_offset_left_rightdistribution_secondresultcongruence = (y) + (p) * pfa_offset_right_rightdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_rightleft. pfa_gap_rightdistribution_rightleft + S (x) = (p)) /\ (((exists pfa_gap_rightdistribution_rightright. pfa_gap_rightdistribution_rightright + S (y) = (p)) /\ ((((exists pfa_gap_rightdistribution_rightresultbound. pfa_gap_rightdistribution_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_rightdistribution_rightresultcongruence pfa_offset_right_rightdistribution_rightresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_rightdistribution_rightresultcongruence = (v) + (p) * pfa_offset_right_rightdistribution_rightresultcongruence))))))))) -> u = v

forall p a. (~((p) = 1) /\ forall pfa_factor_left_add_zero_domain pfa_factor_right_add_zero_domain. (p) = pfa_factor_left_add_zero_domain * pfa_factor_right_add_zero_domain -> pfa_factor_left_add_zero_domain = 1 \/ pfa_factor_right_add_zero_domain = 1) -> (exists pfa_gap_add_zero_bound. pfa_gap_add_zero_bound + S (a) = (p)) -> (((exists pfa_gap_add_zeroleft. pfa_gap_add_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_add_zeroright. pfa_gap_add_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_add_zeroresultbound. pfa_gap_add_zeroresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_add_zeroresultcongruence pfa_offset_right_add_zeroresultcongruence. ((a) + (0)) + (p) * pfa_offset_left_add_zeroresultcongruence = (a) + (p) * pfa_offset_right_add_zeroresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_add_domain pfa_factor_right_zero_add_domain. (p) = pfa_factor_left_zero_add_domain * pfa_factor_right_zero_add_domain -> pfa_factor_left_zero_add_domain = 1 \/ pfa_factor_right_zero_add_domain = 1) -> (exists pfa_gap_zero_add_bound. pfa_gap_zero_add_bound + S (a) = (p)) -> (((exists pfa_gap_zero_addleft. pfa_gap_zero_addleft + S (0) = (p)) /\ (((exists pfa_gap_zero_addright. pfa_gap_zero_addright + S (a) = (p)) /\ ((((exists pfa_gap_zero_addresultbound. pfa_gap_zero_addresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_zero_addresultcongruence pfa_offset_right_zero_addresultcongruence. ((0) + (a)) + (p) * pfa_offset_left_zero_addresultcongruence = (a) + (p) * pfa_offset_right_zero_addresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_one_domain pfa_factor_right_multiply_one_domain. (p) = pfa_factor_left_multiply_one_domain * pfa_factor_right_multiply_one_domain -> pfa_factor_left_multiply_one_domain = 1 \/ pfa_factor_right_multiply_one_domain = 1) -> (exists pfa_gap_multiply_one_bound. pfa_gap_multiply_one_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_oneleft. pfa_gap_multiply_oneleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_oneright. pfa_gap_multiply_oneright + S (1) = (p)) /\ ((((exists pfa_gap_multiply_oneresultbound. pfa_gap_multiply_oneresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_multiply_oneresultcongruence pfa_offset_right_multiply_oneresultcongruence. ((a) * (1)) + (p) * pfa_offset_left_multiply_oneresultcongruence = (a) + (p) * pfa_offset_right_multiply_oneresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_one_multiply_domain pfa_factor_right_one_multiply_domain. (p) = pfa_factor_left_one_multiply_domain * pfa_factor_right_one_multiply_domain -> pfa_factor_left_one_multiply_domain = 1 \/ pfa_factor_right_one_multiply_domain = 1) -> (exists pfa_gap_one_multiply_bound. pfa_gap_one_multiply_bound + S (a) = (p)) -> (((exists pfa_gap_one_multiplyleft. pfa_gap_one_multiplyleft + S (1) = (p)) /\ (((exists pfa_gap_one_multiplyright. pfa_gap_one_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_one_multiplyresultbound. pfa_gap_one_multiplyresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_one_multiplyresultcongruence pfa_offset_right_one_multiplyresultcongruence. ((1) * (a)) + (p) * pfa_offset_left_one_multiplyresultcongruence = (a) + (p) * pfa_offset_right_one_multiplyresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_zero_domain pfa_factor_right_multiply_zero_domain. (p) = pfa_factor_left_multiply_zero_domain * pfa_factor_right_multiply_zero_domain -> pfa_factor_left_multiply_zero_domain = 1 \/ pfa_factor_right_multiply_zero_domain = 1) -> (exists pfa_gap_multiply_zero_bound. pfa_gap_multiply_zero_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_zeroleft. pfa_gap_multiply_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_zeroright. pfa_gap_multiply_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_multiply_zeroresultbound. pfa_gap_multiply_zeroresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_multiply_zeroresultcongruence pfa_offset_right_multiply_zeroresultcongruence. ((a) * (0)) + (p) * pfa_offset_left_multiply_zeroresultcongruence = (0) + (p) * pfa_offset_right_multiply_zeroresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_multiply_domain pfa_factor_right_zero_multiply_domain. (p) = pfa_factor_left_zero_multiply_domain * pfa_factor_right_zero_multiply_domain -> pfa_factor_left_zero_multiply_domain = 1 \/ pfa_factor_right_zero_multiply_domain = 1) -> (exists pfa_gap_zero_multiply_bound. pfa_gap_zero_multiply_bound + S (a) = (p)) -> (((exists pfa_gap_zero_multiplyleft. pfa_gap_zero_multiplyleft + S (0) = (p)) /\ (((exists pfa_gap_zero_multiplyright. pfa_gap_zero_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_zero_multiplyresultbound. pfa_gap_zero_multiplyresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_zero_multiplyresultcongruence pfa_offset_right_zero_multiplyresultcongruence. ((0) * (a)) + (p) * pfa_offset_left_zero_multiplyresultcongruence = (0) + (p) * pfa_offset_right_zero_multiplyresultcongruence)))))))))

forall p a b c z. (((exists pfa_gap_cancel_add_firstleft. pfa_gap_cancel_add_firstleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_firstright. pfa_gap_cancel_add_firstright + S (b) = (p)) /\ ((((exists pfa_gap_cancel_add_firstresultbound. pfa_gap_cancel_add_firstresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_firstresultcongruence pfa_offset_right_cancel_add_firstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_cancel_add_firstresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_firstresultcongruence))))))))) -> (((exists pfa_gap_cancel_add_secondleft. pfa_gap_cancel_add_secondleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_secondright. pfa_gap_cancel_add_secondright + S (c) = (p)) /\ ((((exists pfa_gap_cancel_add_secondresultbound. pfa_gap_cancel_add_secondresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_secondresultcongruence pfa_offset_right_cancel_add_secondresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_cancel_add_secondresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_secondresultcongruence))))))))) -> b = c

forall p a. (~((p) = 1) /\ forall pfa_factor_left_negate_domain pfa_factor_right_negate_domain. (p) = pfa_factor_left_negate_domain * pfa_factor_right_negate_domain -> pfa_factor_left_negate_domain = 1 \/ pfa_factor_right_negate_domain = 1) -> (exists pfa_gap_negate_input. pfa_gap_negate_input + S (a) = (p)) -> exists b. (((exists pfa_gap_negate_existsadditionleft. pfa_gap_negate_existsadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_existsadditionright. pfa_gap_negate_existsadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_existsadditionresultbound. pfa_gap_negate_existsadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_existsadditionresultcongruence pfa_offset_right_negate_existsadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_existsadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_existsadditionresultcongruence)))))))))

forall p a b c. (((exists pfa_gap_negate_firstadditionleft. pfa_gap_negate_firstadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_firstadditionright. pfa_gap_negate_firstadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_firstadditionresultbound. pfa_gap_negate_firstadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_firstadditionresultcongruence pfa_offset_right_negate_firstadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_firstadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_firstadditionresultcongruence))))))))) -> (((exists pfa_gap_negate_secondadditionleft. pfa_gap_negate_secondadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_secondadditionright. pfa_gap_negate_secondadditionright + S (c) = (p)) /\ ((((exists pfa_gap_negate_secondadditionresultbound. pfa_gap_negate_secondadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_secondadditionresultcongruence pfa_offset_right_negate_secondadditionresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_negate_secondadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_secondadditionresultcongruence))))))))) -> b = c

forall p a. (~((p) = 1) /\ forall pfa_factor_left_negate_unique_domain pfa_factor_right_negate_unique_domain. (p) = pfa_factor_left_negate_unique_domain * pfa_factor_right_negate_unique_domain -> pfa_factor_left_negate_unique_domain = 1 \/ pfa_factor_right_negate_unique_domain = 1) -> (exists pfa_gap_negate_unique_input. pfa_gap_negate_unique_input + S (a) = (p)) -> exists b. (((exists pfa_gap_negate_chosenadditionleft. pfa_gap_negate_chosenadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_chosenadditionright. pfa_gap_negate_chosenadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_chosenadditionresultbound. pfa_gap_negate_chosenadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_chosenadditionresultcongruence pfa_offset_right_negate_chosenadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_chosenadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_chosenadditionresultcongruence))))))))) /\ forall c. (((exists pfa_gap_negate_otheradditionleft. pfa_gap_negate_otheradditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_otheradditionright. pfa_gap_negate_otheradditionright + S (c) = (p)) /\ ((((exists pfa_gap_negate_otheradditionresultbound. pfa_gap_negate_otheradditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_otheradditionresultcongruence pfa_offset_right_negate_otheradditionresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_negate_otheradditionresultcongruence = (0) + (p) * pfa_offset_right_negate_otheradditionresultcongruence))))))))) -> c = b

forall p a. (~((p) = 1) /\ forall pfa_factor_left_inverse_domain pfa_factor_right_inverse_domain. (p) = pfa_factor_left_inverse_domain * pfa_factor_right_inverse_domain -> pfa_factor_left_inverse_domain = 1 \/ pfa_factor_right_inverse_domain = 1) -> (exists pfa_gap_inverse_input. pfa_gap_inverse_input + S (a) = (p)) -> ~(a = 0) -> exists b. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_existsmultiplicationleft. pfa_gap_inverse_existsmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_existsmultiplicationright. pfa_gap_inverse_existsmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_existsmultiplicationresultbound. pfa_gap_inverse_existsmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_existsmultiplicationresultcongruence pfa_offset_right_inverse_existsmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_existsmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_existsmultiplicationresultcongruence))))))))))))

forall p a b c. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_firstmultiplicationleft. pfa_gap_inverse_firstmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_firstmultiplicationright. pfa_gap_inverse_firstmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_firstmultiplicationresultbound. pfa_gap_inverse_firstmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_firstmultiplicationresultcongruence pfa_offset_right_inverse_firstmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_firstmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_firstmultiplicationresultcongruence)))))))))))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_secondmultiplicationleft. pfa_gap_inverse_secondmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_secondmultiplicationright. pfa_gap_inverse_secondmultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_inverse_secondmultiplicationresultbound. pfa_gap_inverse_secondmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_secondmultiplicationresultcongruence pfa_offset_right_inverse_secondmultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_inverse_secondmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_secondmultiplicationresultcongruence)))))))))))) -> b = c

forall p a. (~((p) = 1) /\ forall pfa_factor_left_inverse_unique_domain pfa_factor_right_inverse_unique_domain. (p) = pfa_factor_left_inverse_unique_domain * pfa_factor_right_inverse_unique_domain -> pfa_factor_left_inverse_unique_domain = 1 \/ pfa_factor_right_inverse_unique_domain = 1) -> (exists pfa_gap_inverse_unique_input. pfa_gap_inverse_unique_input + S (a) = (p)) -> ~(a = 0) -> exists b. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_chosenmultiplicationleft. pfa_gap_inverse_chosenmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_chosenmultiplicationright. pfa_gap_inverse_chosenmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_chosenmultiplicationresultbound. pfa_gap_inverse_chosenmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_chosenmultiplicationresultcongruence pfa_offset_right_inverse_chosenmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_chosenmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_chosenmultiplicationresultcongruence)))))))))))) /\ forall c. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_othermultiplicationleft. pfa_gap_inverse_othermultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_othermultiplicationright. pfa_gap_inverse_othermultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_inverse_othermultiplicationresultbound. pfa_gap_inverse_othermultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_othermultiplicationresultcongruence pfa_offset_right_inverse_othermultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_inverse_othermultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_othermultiplicationresultcongruence)))))))))))) -> c = b

forall p b. (~((p) = 1) /\ forall pfa_factor_left_zero_inverse_domain pfa_factor_right_zero_inverse_domain. (p) = pfa_factor_left_zero_inverse_domain * pfa_factor_right_zero_inverse_domain -> pfa_factor_left_zero_inverse_domain = 1 \/ pfa_factor_right_zero_inverse_domain = 1) -> ~(((exists pfa_gap_zero_inverseleft. pfa_gap_zero_inverseleft + S (0) = (p)) /\ (((exists pfa_gap_zero_inverseright. pfa_gap_zero_inverseright + S (b) = (p)) /\ ((((exists pfa_gap_zero_inverseresultbound. pfa_gap_zero_inverseresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_inverseresultcongruence pfa_offset_right_zero_inverseresultcongruence. ((0) * (b)) + (p) * pfa_offset_left_zero_inverseresultcongruence = (1) + (p) * pfa_offset_right_zero_inverseresultcongruence)))))))))

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_inverse_output_domain pfa_factor_right_inverse_output_domain. (p) = pfa_factor_left_inverse_output_domain * pfa_factor_right_inverse_output_domain -> pfa_factor_left_inverse_output_domain = 1 \/ pfa_factor_right_inverse_output_domain = 1) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_outputmultiplicationleft. pfa_gap_inverse_outputmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_outputmultiplicationright. pfa_gap_inverse_outputmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_outputmultiplicationresultbound. pfa_gap_inverse_outputmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_outputmultiplicationresultcongruence pfa_offset_right_inverse_outputmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_outputmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_outputmultiplicationresultcongruence)))))))))))) -> ~(b = 0)

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_inverse_symmetric_domain pfa_factor_right_inverse_symmetric_domain. (p) = pfa_factor_left_inverse_symmetric_domain * pfa_factor_right_inverse_symmetric_domain -> pfa_factor_left_inverse_symmetric_domain = 1 \/ pfa_factor_right_inverse_symmetric_domain = 1) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_symmetric_sourcemultiplicationleft. pfa_gap_inverse_symmetric_sourcemultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_symmetric_sourcemultiplicationright. pfa_gap_inverse_symmetric_sourcemultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_symmetric_sourcemultiplicationresultbound. pfa_gap_inverse_symmetric_sourcemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_symmetric_sourcemultiplicationresultcongruence pfa_offset_right_inverse_symmetric_sourcemultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_symmetric_sourcemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_symmetric_sourcemultiplicationresultcongruence)))))))))))) -> (((~((b) = 0)) /\ ((((exists pfa_gap_inverse_symmetric_targetmultiplicationleft. pfa_gap_inverse_symmetric_targetmultiplicationleft + S (b) = (p)) /\ (((exists pfa_gap_inverse_symmetric_targetmultiplicationright. pfa_gap_inverse_symmetric_targetmultiplicationright + S (a) = (p)) /\ ((((exists pfa_gap_inverse_symmetric_targetmultiplicationresultbound. pfa_gap_inverse_symmetric_targetmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_symmetric_targetmultiplicationresultcongruence pfa_offset_right_inverse_symmetric_targetmultiplicationresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_inverse_symmetric_targetmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_symmetric_targetmultiplicationresultcongruence))))))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_coprime_domain pfa_factor_right_coprime_domain. (p) = pfa_factor_left_coprime_domain * pfa_factor_right_coprime_domain -> pfa_factor_left_coprime_domain = 1 \/ pfa_factor_right_coprime_domain = 1) -> (exists pfa_gap_coprime_bound. pfa_gap_coprime_bound + S (a) = (p)) -> ~(a = 0) -> (forall pfa_divisor_coprime_result. (exists pfa_left_factor_coprime_result. (a) = pfa_divisor_coprime_result * pfa_left_factor_coprime_result) -> (exists pfa_right_factor_coprime_result. (p) = pfa_divisor_coprime_result * pfa_right_factor_coprime_result) -> pfa_divisor_coprime_result = 1)

forall p a b c z. (~((p) = 1) /\ forall pfa_factor_left_cancel_mul_domain pfa_factor_right_cancel_mul_domain. (p) = pfa_factor_left_cancel_mul_domain * pfa_factor_right_cancel_mul_domain -> pfa_factor_left_cancel_mul_domain = 1 \/ pfa_factor_right_cancel_mul_domain = 1) -> ~(a = 0) -> (((exists pfa_gap_cancel_mul_firstleft. pfa_gap_cancel_mul_firstleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_mul_firstright. pfa_gap_cancel_mul_firstright + S (b) = (p)) /\ ((((exists pfa_gap_cancel_mul_firstresultbound. pfa_gap_cancel_mul_firstresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_mul_firstresultcongruence pfa_offset_right_cancel_mul_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_cancel_mul_firstresultcongruence = (z) + (p) * pfa_offset_right_cancel_mul_firstresultcongruence))))))))) -> (((exists pfa_gap_cancel_mul_secondleft. pfa_gap_cancel_mul_secondleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_mul_secondright. pfa_gap_cancel_mul_secondright + S (c) = (p)) /\ ((((exists pfa_gap_cancel_mul_secondresultbound. pfa_gap_cancel_mul_secondresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_mul_secondresultcongruence pfa_offset_right_cancel_mul_secondresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_cancel_mul_secondresultcongruence = (z) + (p) * pfa_offset_right_cancel_mul_secondresultcongruence))))))))) -> b = c

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_no_zero_divisors_domain pfa_factor_right_no_zero_divisors_domain. (p) = pfa_factor_left_no_zero_divisors_domain * pfa_factor_right_no_zero_divisors_domain -> pfa_factor_left_no_zero_divisors_domain = 1 \/ pfa_factor_right_no_zero_divisors_domain = 1) -> (((exists pfa_gap_no_zero_divisors_productleft. pfa_gap_no_zero_divisors_productleft + S (a) = (p)) /\ (((exists pfa_gap_no_zero_divisors_productright. pfa_gap_no_zero_divisors_productright + S (b) = (p)) /\ ((((exists pfa_gap_no_zero_divisors_productresultbound. pfa_gap_no_zero_divisors_productresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_no_zero_divisors_productresultcongruence pfa_offset_right_no_zero_divisors_productresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_no_zero_divisors_productresultcongruence = (0) + (p) * pfa_offset_right_no_zero_divisors_productresultcongruence))))))))) -> a = 0 \/ b = 0

forall p a b x y z. (((exists pfa_gap_addnatural_leftbound. pfa_gap_addnatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_addnatural_leftcongruence pfa_offset_right_addnatural_leftcongruence. (a) + (p) * pfa_offset_left_addnatural_leftcongruence = (x) + (p) * pfa_offset_right_addnatural_leftcongruence)))) -> (((exists pfa_gap_addnatural_rightbound. pfa_gap_addnatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_addnatural_rightcongruence pfa_offset_right_addnatural_rightcongruence. (b) + (p) * pfa_offset_left_addnatural_rightcongruence = (y) + (p) * pfa_offset_right_addnatural_rightcongruence)))) -> (((exists pfa_gap_addnatural_resultleft. pfa_gap_addnatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_addnatural_resultright. pfa_gap_addnatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_addnatural_resultresultbound. pfa_gap_addnatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_addnatural_resultresultcongruence pfa_offset_right_addnatural_resultresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_addnatural_resultresultcongruence = (z) + (p) * pfa_offset_right_addnatural_resultresultcongruence))))))))) -> (((exists pfa_gap_addnatural_homomorphismbound. pfa_gap_addnatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_addnatural_homomorphismcongruence pfa_offset_right_addnatural_homomorphismcongruence. (a + b) + (p) * pfa_offset_left_addnatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_addnatural_homomorphismcongruence))))

forall p a b x y z. (((exists pfa_gap_multiplynatural_leftbound. pfa_gap_multiplynatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplynatural_leftcongruence pfa_offset_right_multiplynatural_leftcongruence. (a) + (p) * pfa_offset_left_multiplynatural_leftcongruence = (x) + (p) * pfa_offset_right_multiplynatural_leftcongruence)))) -> (((exists pfa_gap_multiplynatural_rightbound. pfa_gap_multiplynatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplynatural_rightcongruence pfa_offset_right_multiplynatural_rightcongruence. (b) + (p) * pfa_offset_left_multiplynatural_rightcongruence = (y) + (p) * pfa_offset_right_multiplynatural_rightcongruence)))) -> (((exists pfa_gap_multiplynatural_resultleft. pfa_gap_multiplynatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_multiplynatural_resultright. pfa_gap_multiplynatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_multiplynatural_resultresultbound. pfa_gap_multiplynatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_resultresultcongruence pfa_offset_right_multiplynatural_resultresultcongruence. ((x) * (y)) + (p) * pfa_offset_left_multiplynatural_resultresultcongruence = (z) + (p) * pfa_offset_right_multiplynatural_resultresultcongruence))))))))) -> (((exists pfa_gap_multiplynatural_homomorphismbound. pfa_gap_multiplynatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_homomorphismcongruence pfa_offset_right_multiplynatural_homomorphismcongruence. (a * b) + (p) * pfa_offset_left_multiplynatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_multiplynatural_homomorphismcongruence))))

forall p n. (exists pfa_gap_small_nonzero_bound. pfa_gap_small_nonzero_bound + S (n) = (p)) -> ~(n = 0) -> ~(((exists pfa_gap_small_nonzerobound. pfa_gap_small_nonzerobound + S (0) = (p)) /\ ((exists pfa_offset_left_small_nonzerocongruence pfa_offset_right_small_nonzerocongruence. (n) + (p) * pfa_offset_left_small_nonzerocongruence = (0) + (p) * pfa_offset_right_small_nonzerocongruence))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_complete_laws_domain pfa_factor_right_complete_laws_domain. (p) = pfa_factor_left_complete_laws_domain * pfa_factor_right_complete_laws_domain -> pfa_factor_left_complete_laws_domain = 1 \/ pfa_factor_right_complete_laws_domain = 1) -> (((exists pfa_gap_complete_lawszero. pfa_gap_complete_lawszero + S (0) = (p)) /\ (((exists pfa_gap_complete_lawsone. pfa_gap_complete_lawsone + S (1) = (p)) /\ (((~(0 = 1)) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws. (exists pfa_gap_complete_lawsaddleft. pfa_gap_complete_lawsaddleft + S (pfa_law_a_complete_laws) = (p)) -> (exists pfa_gap_complete_lawsaddright. pfa_gap_complete_lawsaddright + S (pfa_law_b_complete_laws) = (p)) -> exists pfa_law_c_complete_laws. (((exists pfa_gap_complete_lawsaddchosenleft. pfa_gap_complete_lawsaddchosenleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddchosenright. pfa_gap_complete_lawsaddchosenright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddchosenresultbound. pfa_gap_complete_lawsaddchosenresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddchosenresultcongruence pfa_offset_right_complete_lawsaddchosenresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddchosenresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsaddchosenresultcongruence))))))))) /\ forall pfa_law_d_complete_laws. (((exists pfa_gap_complete_lawsaddotherleft. pfa_gap_complete_lawsaddotherleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddotherright. pfa_gap_complete_lawsaddotherright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddotherresultbound. pfa_gap_complete_lawsaddotherresultbound + S (pfa_law_d_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddotherresultcongruence pfa_offset_right_complete_lawsaddotherresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddotherresultcongruence = (pfa_law_d_complete_laws) + (p) * pfa_offset_right_complete_lawsaddotherresultcongruence))))))))) -> pfa_law_d_complete_laws = pfa_law_c_complete_laws) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws. (((exists pfa_gap_complete_lawsaddcomm_firstleft. pfa_gap_complete_lawsaddcomm_firstleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddcomm_firstright. pfa_gap_complete_lawsaddcomm_firstright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddcomm_firstresultbound. pfa_gap_complete_lawsaddcomm_firstresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddcomm_firstresultcongruence pfa_offset_right_complete_lawsaddcomm_firstresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddcomm_firstresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsaddcomm_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsaddcomm_secondleft. pfa_gap_complete_lawsaddcomm_secondleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddcomm_secondright. pfa_gap_complete_lawsaddcomm_secondright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddcomm_secondresultbound. pfa_gap_complete_lawsaddcomm_secondresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddcomm_secondresultcongruence pfa_offset_right_complete_lawsaddcomm_secondresultcongruence. ((pfa_law_b_complete_laws) + (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddcomm_secondresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsaddcomm_secondresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws pfa_law_x_complete_laws pfa_law_y_complete_laws pfa_law_u_complete_laws pfa_law_v_complete_laws. (((exists pfa_gap_complete_lawsaddassoc_firstleft. pfa_gap_complete_lawsaddassoc_firstleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddassoc_firstright. pfa_gap_complete_lawsaddassoc_firstright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddassoc_firstresultbound. pfa_gap_complete_lawsaddassoc_firstresultbound + S (pfa_law_x_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddassoc_firstresultcongruence pfa_offset_right_complete_lawsaddassoc_firstresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddassoc_firstresultcongruence = (pfa_law_x_complete_laws) + (p) * pfa_offset_right_complete_lawsaddassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsaddassoc_leftleft. pfa_gap_complete_lawsaddassoc_leftleft + S (pfa_law_x_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddassoc_leftright. pfa_gap_complete_lawsaddassoc_leftright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddassoc_leftresultbound. pfa_gap_complete_lawsaddassoc_leftresultbound + S (pfa_law_u_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddassoc_leftresultcongruence pfa_offset_right_complete_lawsaddassoc_leftresultcongruence. ((pfa_law_x_complete_laws) + (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddassoc_leftresultcongruence = (pfa_law_u_complete_laws) + (p) * pfa_offset_right_complete_lawsaddassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsaddassoc_secondleft. pfa_gap_complete_lawsaddassoc_secondleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddassoc_secondright. pfa_gap_complete_lawsaddassoc_secondright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddassoc_secondresultbound. pfa_gap_complete_lawsaddassoc_secondresultbound + S (pfa_law_y_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddassoc_secondresultcongruence pfa_offset_right_complete_lawsaddassoc_secondresultcongruence. ((pfa_law_b_complete_laws) + (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddassoc_secondresultcongruence = (pfa_law_y_complete_laws) + (p) * pfa_offset_right_complete_lawsaddassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsaddassoc_rightleft. pfa_gap_complete_lawsaddassoc_rightleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsaddassoc_rightright. pfa_gap_complete_lawsaddassoc_rightright + S (pfa_law_y_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsaddassoc_rightresultbound. pfa_gap_complete_lawsaddassoc_rightresultbound + S (pfa_law_v_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsaddassoc_rightresultcongruence pfa_offset_right_complete_lawsaddassoc_rightresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_y_complete_laws)) + (p) * pfa_offset_left_complete_lawsaddassoc_rightresultcongruence = (pfa_law_v_complete_laws) + (p) * pfa_offset_right_complete_lawsaddassoc_rightresultcongruence))))))))) -> pfa_law_u_complete_laws = pfa_law_v_complete_laws) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws. (exists pfa_gap_complete_lawsmultiplyleft. pfa_gap_complete_lawsmultiplyleft + S (pfa_law_a_complete_laws) = (p)) -> (exists pfa_gap_complete_lawsmultiplyright. pfa_gap_complete_lawsmultiplyright + S (pfa_law_b_complete_laws) = (p)) -> exists pfa_law_c_complete_laws. (((exists pfa_gap_complete_lawsmultiplychosenleft. pfa_gap_complete_lawsmultiplychosenleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplychosenright. pfa_gap_complete_lawsmultiplychosenright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplychosenresultbound. pfa_gap_complete_lawsmultiplychosenresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplychosenresultcongruence pfa_offset_right_complete_lawsmultiplychosenresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplychosenresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplychosenresultcongruence))))))))) /\ forall pfa_law_d_complete_laws. (((exists pfa_gap_complete_lawsmultiplyotherleft. pfa_gap_complete_lawsmultiplyotherleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplyotherright. pfa_gap_complete_lawsmultiplyotherright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplyotherresultbound. pfa_gap_complete_lawsmultiplyotherresultbound + S (pfa_law_d_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplyotherresultcongruence pfa_offset_right_complete_lawsmultiplyotherresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplyotherresultcongruence = (pfa_law_d_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplyotherresultcongruence))))))))) -> pfa_law_d_complete_laws = pfa_law_c_complete_laws) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws. (((exists pfa_gap_complete_lawsmultiplycomm_firstleft. pfa_gap_complete_lawsmultiplycomm_firstleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplycomm_firstright. pfa_gap_complete_lawsmultiplycomm_firstright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplycomm_firstresultbound. pfa_gap_complete_lawsmultiplycomm_firstresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplycomm_firstresultcongruence pfa_offset_right_complete_lawsmultiplycomm_firstresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplycomm_firstresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplycomm_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsmultiplycomm_secondleft. pfa_gap_complete_lawsmultiplycomm_secondleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplycomm_secondright. pfa_gap_complete_lawsmultiplycomm_secondright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplycomm_secondresultbound. pfa_gap_complete_lawsmultiplycomm_secondresultbound + S (pfa_law_c_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplycomm_secondresultcongruence pfa_offset_right_complete_lawsmultiplycomm_secondresultcongruence. ((pfa_law_b_complete_laws) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplycomm_secondresultcongruence = (pfa_law_c_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplycomm_secondresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws pfa_law_x_complete_laws pfa_law_y_complete_laws pfa_law_u_complete_laws pfa_law_v_complete_laws. (((exists pfa_gap_complete_lawsmultiplyassoc_firstleft. pfa_gap_complete_lawsmultiplyassoc_firstleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplyassoc_firstright. pfa_gap_complete_lawsmultiplyassoc_firstright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplyassoc_firstresultbound. pfa_gap_complete_lawsmultiplyassoc_firstresultbound + S (pfa_law_x_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplyassoc_firstresultcongruence pfa_offset_right_complete_lawsmultiplyassoc_firstresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplyassoc_firstresultcongruence = (pfa_law_x_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplyassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsmultiplyassoc_leftleft. pfa_gap_complete_lawsmultiplyassoc_leftleft + S (pfa_law_x_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplyassoc_leftright. pfa_gap_complete_lawsmultiplyassoc_leftright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplyassoc_leftresultbound. pfa_gap_complete_lawsmultiplyassoc_leftresultbound + S (pfa_law_u_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplyassoc_leftresultcongruence pfa_offset_right_complete_lawsmultiplyassoc_leftresultcongruence. ((pfa_law_x_complete_laws) * (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplyassoc_leftresultcongruence = (pfa_law_u_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplyassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsmultiplyassoc_secondleft. pfa_gap_complete_lawsmultiplyassoc_secondleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplyassoc_secondright. pfa_gap_complete_lawsmultiplyassoc_secondright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplyassoc_secondresultbound. pfa_gap_complete_lawsmultiplyassoc_secondresultbound + S (pfa_law_y_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplyassoc_secondresultcongruence pfa_offset_right_complete_lawsmultiplyassoc_secondresultcongruence. ((pfa_law_b_complete_laws) * (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplyassoc_secondresultcongruence = (pfa_law_y_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplyassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsmultiplyassoc_rightleft. pfa_gap_complete_lawsmultiplyassoc_rightleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiplyassoc_rightright. pfa_gap_complete_lawsmultiplyassoc_rightright + S (pfa_law_y_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiplyassoc_rightresultbound. pfa_gap_complete_lawsmultiplyassoc_rightresultbound + S (pfa_law_v_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiplyassoc_rightresultcongruence pfa_offset_right_complete_lawsmultiplyassoc_rightresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_y_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiplyassoc_rightresultcongruence = (pfa_law_v_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiplyassoc_rightresultcongruence))))))))) -> pfa_law_u_complete_laws = pfa_law_v_complete_laws) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws pfa_law_s_complete_laws pfa_law_x_complete_laws pfa_law_y_complete_laws pfa_law_u_complete_laws pfa_law_v_complete_laws. (((exists pfa_gap_complete_lawsleftdistribution_sumleft. pfa_gap_complete_lawsleftdistribution_sumleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsleftdistribution_sumright. pfa_gap_complete_lawsleftdistribution_sumright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsleftdistribution_sumresultbound. pfa_gap_complete_lawsleftdistribution_sumresultbound + S (pfa_law_s_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsleftdistribution_sumresultcongruence pfa_offset_right_complete_lawsleftdistribution_sumresultcongruence. ((pfa_law_b_complete_laws) + (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsleftdistribution_sumresultcongruence = (pfa_law_s_complete_laws) + (p) * pfa_offset_right_complete_lawsleftdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsleftdistribution_leftleft. pfa_gap_complete_lawsleftdistribution_leftleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsleftdistribution_leftright. pfa_gap_complete_lawsleftdistribution_leftright + S (pfa_law_s_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsleftdistribution_leftresultbound. pfa_gap_complete_lawsleftdistribution_leftresultbound + S (pfa_law_u_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsleftdistribution_leftresultcongruence pfa_offset_right_complete_lawsleftdistribution_leftresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_s_complete_laws)) + (p) * pfa_offset_left_complete_lawsleftdistribution_leftresultcongruence = (pfa_law_u_complete_laws) + (p) * pfa_offset_right_complete_lawsleftdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsleftdistribution_firstleft. pfa_gap_complete_lawsleftdistribution_firstleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsleftdistribution_firstright. pfa_gap_complete_lawsleftdistribution_firstright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsleftdistribution_firstresultbound. pfa_gap_complete_lawsleftdistribution_firstresultbound + S (pfa_law_x_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsleftdistribution_firstresultcongruence pfa_offset_right_complete_lawsleftdistribution_firstresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsleftdistribution_firstresultcongruence = (pfa_law_x_complete_laws) + (p) * pfa_offset_right_complete_lawsleftdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsleftdistribution_secondleft. pfa_gap_complete_lawsleftdistribution_secondleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsleftdistribution_secondright. pfa_gap_complete_lawsleftdistribution_secondright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsleftdistribution_secondresultbound. pfa_gap_complete_lawsleftdistribution_secondresultbound + S (pfa_law_y_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsleftdistribution_secondresultcongruence pfa_offset_right_complete_lawsleftdistribution_secondresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsleftdistribution_secondresultcongruence = (pfa_law_y_complete_laws) + (p) * pfa_offset_right_complete_lawsleftdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsleftdistribution_rightleft. pfa_gap_complete_lawsleftdistribution_rightleft + S (pfa_law_x_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsleftdistribution_rightright. pfa_gap_complete_lawsleftdistribution_rightright + S (pfa_law_y_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsleftdistribution_rightresultbound. pfa_gap_complete_lawsleftdistribution_rightresultbound + S (pfa_law_v_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsleftdistribution_rightresultcongruence pfa_offset_right_complete_lawsleftdistribution_rightresultcongruence. ((pfa_law_x_complete_laws) + (pfa_law_y_complete_laws)) + (p) * pfa_offset_left_complete_lawsleftdistribution_rightresultcongruence = (pfa_law_v_complete_laws) + (p) * pfa_offset_right_complete_lawsleftdistribution_rightresultcongruence))))))))) -> pfa_law_u_complete_laws = pfa_law_v_complete_laws) /\ (((forall pfa_law_a_complete_laws pfa_law_b_complete_laws pfa_law_c_complete_laws pfa_law_s_complete_laws pfa_law_x_complete_laws pfa_law_y_complete_laws pfa_law_u_complete_laws pfa_law_v_complete_laws. (((exists pfa_gap_complete_lawsrightdistribution_sumleft. pfa_gap_complete_lawsrightdistribution_sumleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsrightdistribution_sumright. pfa_gap_complete_lawsrightdistribution_sumright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsrightdistribution_sumresultbound. pfa_gap_complete_lawsrightdistribution_sumresultbound + S (pfa_law_s_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsrightdistribution_sumresultcongruence pfa_offset_right_complete_lawsrightdistribution_sumresultcongruence. ((pfa_law_b_complete_laws) + (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsrightdistribution_sumresultcongruence = (pfa_law_s_complete_laws) + (p) * pfa_offset_right_complete_lawsrightdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsrightdistribution_leftleft. pfa_gap_complete_lawsrightdistribution_leftleft + S (pfa_law_s_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsrightdistribution_leftright. pfa_gap_complete_lawsrightdistribution_leftright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsrightdistribution_leftresultbound. pfa_gap_complete_lawsrightdistribution_leftresultbound + S (pfa_law_u_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsrightdistribution_leftresultcongruence pfa_offset_right_complete_lawsrightdistribution_leftresultcongruence. ((pfa_law_s_complete_laws) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsrightdistribution_leftresultcongruence = (pfa_law_u_complete_laws) + (p) * pfa_offset_right_complete_lawsrightdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsrightdistribution_firstleft. pfa_gap_complete_lawsrightdistribution_firstleft + S (pfa_law_b_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsrightdistribution_firstright. pfa_gap_complete_lawsrightdistribution_firstright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsrightdistribution_firstresultbound. pfa_gap_complete_lawsrightdistribution_firstresultbound + S (pfa_law_x_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsrightdistribution_firstresultcongruence pfa_offset_right_complete_lawsrightdistribution_firstresultcongruence. ((pfa_law_b_complete_laws) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsrightdistribution_firstresultcongruence = (pfa_law_x_complete_laws) + (p) * pfa_offset_right_complete_lawsrightdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsrightdistribution_secondleft. pfa_gap_complete_lawsrightdistribution_secondleft + S (pfa_law_c_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsrightdistribution_secondright. pfa_gap_complete_lawsrightdistribution_secondright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsrightdistribution_secondresultbound. pfa_gap_complete_lawsrightdistribution_secondresultbound + S (pfa_law_y_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsrightdistribution_secondresultcongruence pfa_offset_right_complete_lawsrightdistribution_secondresultcongruence. ((pfa_law_c_complete_laws) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsrightdistribution_secondresultcongruence = (pfa_law_y_complete_laws) + (p) * pfa_offset_right_complete_lawsrightdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_complete_lawsrightdistribution_rightleft. pfa_gap_complete_lawsrightdistribution_rightleft + S (pfa_law_x_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsrightdistribution_rightright. pfa_gap_complete_lawsrightdistribution_rightright + S (pfa_law_y_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsrightdistribution_rightresultbound. pfa_gap_complete_lawsrightdistribution_rightresultbound + S (pfa_law_v_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsrightdistribution_rightresultcongruence pfa_offset_right_complete_lawsrightdistribution_rightresultcongruence. ((pfa_law_x_complete_laws) + (pfa_law_y_complete_laws)) + (p) * pfa_offset_left_complete_lawsrightdistribution_rightresultcongruence = (pfa_law_v_complete_laws) + (p) * pfa_offset_right_complete_lawsrightdistribution_rightresultcongruence))))))))) -> pfa_law_u_complete_laws = pfa_law_v_complete_laws) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsadd_zero_rightinput. pfa_gap_complete_lawsadd_zero_rightinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsadd_zero_rightleft. pfa_gap_complete_lawsadd_zero_rightleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsadd_zero_rightright. pfa_gap_complete_lawsadd_zero_rightright + S (0) = (p)) /\ ((((exists pfa_gap_complete_lawsadd_zero_rightresultbound. pfa_gap_complete_lawsadd_zero_rightresultbound + S (pfa_law_a_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsadd_zero_rightresultcongruence pfa_offset_right_complete_lawsadd_zero_rightresultcongruence. ((pfa_law_a_complete_laws) + (0)) + (p) * pfa_offset_left_complete_lawsadd_zero_rightresultcongruence = (pfa_law_a_complete_laws) + (p) * pfa_offset_right_complete_lawsadd_zero_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsadd_zero_leftinput. pfa_gap_complete_lawsadd_zero_leftinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsadd_zero_leftleft. pfa_gap_complete_lawsadd_zero_leftleft + S (0) = (p)) /\ (((exists pfa_gap_complete_lawsadd_zero_leftright. pfa_gap_complete_lawsadd_zero_leftright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsadd_zero_leftresultbound. pfa_gap_complete_lawsadd_zero_leftresultbound + S (pfa_law_a_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsadd_zero_leftresultcongruence pfa_offset_right_complete_lawsadd_zero_leftresultcongruence. ((0) + (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsadd_zero_leftresultcongruence = (pfa_law_a_complete_laws) + (p) * pfa_offset_right_complete_lawsadd_zero_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsmultiply_one_rightinput. pfa_gap_complete_lawsmultiply_one_rightinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsmultiply_one_rightleft. pfa_gap_complete_lawsmultiply_one_rightleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiply_one_rightright. pfa_gap_complete_lawsmultiply_one_rightright + S (1) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiply_one_rightresultbound. pfa_gap_complete_lawsmultiply_one_rightresultbound + S (pfa_law_a_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiply_one_rightresultcongruence pfa_offset_right_complete_lawsmultiply_one_rightresultcongruence. ((pfa_law_a_complete_laws) * (1)) + (p) * pfa_offset_left_complete_lawsmultiply_one_rightresultcongruence = (pfa_law_a_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiply_one_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsmultiply_one_leftinput. pfa_gap_complete_lawsmultiply_one_leftinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsmultiply_one_leftleft. pfa_gap_complete_lawsmultiply_one_leftleft + S (1) = (p)) /\ (((exists pfa_gap_complete_lawsmultiply_one_leftright. pfa_gap_complete_lawsmultiply_one_leftright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiply_one_leftresultbound. pfa_gap_complete_lawsmultiply_one_leftresultbound + S (pfa_law_a_complete_laws) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiply_one_leftresultcongruence pfa_offset_right_complete_lawsmultiply_one_leftresultcongruence. ((1) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiply_one_leftresultcongruence = (pfa_law_a_complete_laws) + (p) * pfa_offset_right_complete_lawsmultiply_one_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsmultiply_zero_rightinput. pfa_gap_complete_lawsmultiply_zero_rightinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsmultiply_zero_rightleft. pfa_gap_complete_lawsmultiply_zero_rightleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsmultiply_zero_rightright. pfa_gap_complete_lawsmultiply_zero_rightright + S (0) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiply_zero_rightresultbound. pfa_gap_complete_lawsmultiply_zero_rightresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiply_zero_rightresultcongruence pfa_offset_right_complete_lawsmultiply_zero_rightresultcongruence. ((pfa_law_a_complete_laws) * (0)) + (p) * pfa_offset_left_complete_lawsmultiply_zero_rightresultcongruence = (0) + (p) * pfa_offset_right_complete_lawsmultiply_zero_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsmultiply_zero_leftinput. pfa_gap_complete_lawsmultiply_zero_leftinput + S (pfa_law_a_complete_laws) = (p)) -> (((exists pfa_gap_complete_lawsmultiply_zero_leftleft. pfa_gap_complete_lawsmultiply_zero_leftleft + S (0) = (p)) /\ (((exists pfa_gap_complete_lawsmultiply_zero_leftright. pfa_gap_complete_lawsmultiply_zero_leftright + S (pfa_law_a_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsmultiply_zero_leftresultbound. pfa_gap_complete_lawsmultiply_zero_leftresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_complete_lawsmultiply_zero_leftresultcongruence pfa_offset_right_complete_lawsmultiply_zero_leftresultcongruence. ((0) * (pfa_law_a_complete_laws)) + (p) * pfa_offset_left_complete_lawsmultiply_zero_leftresultcongruence = (0) + (p) * pfa_offset_right_complete_lawsmultiply_zero_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsnegateinput. pfa_gap_complete_lawsnegateinput + S (pfa_law_a_complete_laws) = (p)) -> exists pfa_law_b_complete_laws. (((exists pfa_gap_complete_lawsnegatechosenadditionleft. pfa_gap_complete_lawsnegatechosenadditionleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsnegatechosenadditionright. pfa_gap_complete_lawsnegatechosenadditionright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsnegatechosenadditionresultbound. pfa_gap_complete_lawsnegatechosenadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_complete_lawsnegatechosenadditionresultcongruence pfa_offset_right_complete_lawsnegatechosenadditionresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsnegatechosenadditionresultcongruence = (0) + (p) * pfa_offset_right_complete_lawsnegatechosenadditionresultcongruence))))))))) /\ forall pfa_law_c_complete_laws. (((exists pfa_gap_complete_lawsnegateotheradditionleft. pfa_gap_complete_lawsnegateotheradditionleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsnegateotheradditionright. pfa_gap_complete_lawsnegateotheradditionright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsnegateotheradditionresultbound. pfa_gap_complete_lawsnegateotheradditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_complete_lawsnegateotheradditionresultcongruence pfa_offset_right_complete_lawsnegateotheradditionresultcongruence. ((pfa_law_a_complete_laws) + (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsnegateotheradditionresultcongruence = (0) + (p) * pfa_offset_right_complete_lawsnegateotheradditionresultcongruence))))))))) -> pfa_law_c_complete_laws = pfa_law_b_complete_laws) /\ (((forall pfa_law_a_complete_laws. (exists pfa_gap_complete_lawsinverseinput. pfa_gap_complete_lawsinverseinput + S (pfa_law_a_complete_laws) = (p)) -> ~(pfa_law_a_complete_laws = 0) -> exists pfa_law_b_complete_laws. (((~((pfa_law_a_complete_laws) = 0)) /\ ((((exists pfa_gap_complete_lawsinversechosenmultiplicationleft. pfa_gap_complete_lawsinversechosenmultiplicationleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsinversechosenmultiplicationright. pfa_gap_complete_lawsinversechosenmultiplicationright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsinversechosenmultiplicationresultbound. pfa_gap_complete_lawsinversechosenmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_complete_lawsinversechosenmultiplicationresultcongruence pfa_offset_right_complete_lawsinversechosenmultiplicationresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsinversechosenmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_complete_lawsinversechosenmultiplicationresultcongruence)))))))))))) /\ forall pfa_law_c_complete_laws. (((~((pfa_law_a_complete_laws) = 0)) /\ ((((exists pfa_gap_complete_lawsinverseothermultiplicationleft. pfa_gap_complete_lawsinverseothermultiplicationleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsinverseothermultiplicationright. pfa_gap_complete_lawsinverseothermultiplicationright + S (pfa_law_c_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsinverseothermultiplicationresultbound. pfa_gap_complete_lawsinverseothermultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_complete_lawsinverseothermultiplicationresultcongruence pfa_offset_right_complete_lawsinverseothermultiplicationresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_c_complete_laws)) + (p) * pfa_offset_left_complete_lawsinverseothermultiplicationresultcongruence = (1) + (p) * pfa_offset_right_complete_lawsinverseothermultiplicationresultcongruence)))))))))))) -> pfa_law_c_complete_laws = pfa_law_b_complete_laws) /\ ((forall pfa_law_a_complete_laws pfa_law_b_complete_laws. (((exists pfa_gap_complete_lawsnozeroleft. pfa_gap_complete_lawsnozeroleft + S (pfa_law_a_complete_laws) = (p)) /\ (((exists pfa_gap_complete_lawsnozeroright. pfa_gap_complete_lawsnozeroright + S (pfa_law_b_complete_laws) = (p)) /\ ((((exists pfa_gap_complete_lawsnozeroresultbound. pfa_gap_complete_lawsnozeroresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_complete_lawsnozeroresultcongruence pfa_offset_right_complete_lawsnozeroresultcongruence. ((pfa_law_a_complete_laws) * (pfa_law_b_complete_laws)) + (p) * pfa_offset_left_complete_lawsnozeroresultcongruence = (0) + (p) * pfa_offset_right_complete_lawsnozeroresultcongruence))))))))) -> pfa_law_a_complete_laws = 0 \/ pfa_law_b_complete_laws = 0))))))))))))))))))))))))))))))))))))))))

forall p i. (~((p) = 1) /\ forall pfa_factor_left_addgrid_domain pfa_factor_right_addgrid_domain. (p) = pfa_factor_left_addgrid_domain * pfa_factor_right_addgrid_domain -> pfa_factor_left_addgrid_domain = 1 \/ pfa_factor_right_addgrid_domain = 1) -> (exists pfa_gap_addgrid_bound. pfa_gap_addgrid_bound + S (i) = (p*p)) -> exists v. (exists pft_row_addgrid_result pft_column_addgrid_result. (((i) = pft_row_addgrid_result * (p) + pft_column_addgrid_result) /\ ((((exists pfa_gap_addgrid_resultoperationleft. pfa_gap_addgrid_resultoperationleft + S (pft_row_addgrid_result) = (p)) /\ (((exists pfa_gap_addgrid_resultoperationright. pfa_gap_addgrid_resultoperationright + S (pft_column_addgrid_result) = (p)) /\ ((((exists pfa_gap_addgrid_resultoperationresultbound. pfa_gap_addgrid_resultoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addgrid_resultoperationresultcongruence pfa_offset_right_addgrid_resultoperationresultcongruence. ((pft_row_addgrid_result) + (pft_column_addgrid_result)) + (p) * pfa_offset_left_addgrid_resultoperationresultcongruence = (v) + (p) * pfa_offset_right_addgrid_resultoperationresultcongruence))))))))))))

forall p i. (~((p) = 1) /\ forall pfa_factor_left_multiplygrid_domain pfa_factor_right_multiplygrid_domain. (p) = pfa_factor_left_multiplygrid_domain * pfa_factor_right_multiplygrid_domain -> pfa_factor_left_multiplygrid_domain = 1 \/ pfa_factor_right_multiplygrid_domain = 1) -> (exists pfa_gap_multiplygrid_bound. pfa_gap_multiplygrid_bound + S (i) = (p*p)) -> exists v. (exists pft_row_multiplygrid_result pft_column_multiplygrid_result. (((i) = pft_row_multiplygrid_result * (p) + pft_column_multiplygrid_result) /\ ((((exists pfa_gap_multiplygrid_resultoperationleft. pfa_gap_multiplygrid_resultoperationleft + S (pft_row_multiplygrid_result) = (p)) /\ (((exists pfa_gap_multiplygrid_resultoperationright. pfa_gap_multiplygrid_resultoperationright + S (pft_column_multiplygrid_result) = (p)) /\ ((((exists pfa_gap_multiplygrid_resultoperationresultbound. pfa_gap_multiplygrid_resultoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplygrid_resultoperationresultcongruence pfa_offset_right_multiplygrid_resultoperationresultcongruence. ((pft_row_multiplygrid_result) * (pft_column_multiplygrid_result)) + (p) * pfa_offset_left_multiplygrid_resultoperationresultcongruence = (v) + (p) * pfa_offset_right_multiplygrid_resultoperationresultcongruence))))))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_extended_domain pfa_factor_right_zero_extended_domain. (p) = pfa_factor_left_zero_extended_domain * pfa_factor_right_zero_extended_domain -> pfa_factor_left_zero_extended_domain = 1 \/ pfa_factor_right_zero_extended_domain = 1) -> (exists pfa_gap_zero_extended_bound. pfa_gap_zero_extended_bound + S (a) = (p)) -> exists b. (((exists pfa_gap_zero_extended_resultinput. pfa_gap_zero_extended_resultinput + S (a) = (p)) /\ (((exists pfa_gap_zero_extended_resultoutput. pfa_gap_zero_extended_resultoutput + S (b) = (p)) /\ ((((a) = 0 /\ (b) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_zero_extended_resultnonzeromultiplicationleft. pfa_gap_zero_extended_resultnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_zero_extended_resultnonzeromultiplicationright. pfa_gap_zero_extended_resultnonzeromultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_zero_extended_resultnonzeromultiplicationresultbound. pfa_gap_zero_extended_resultnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_extended_resultnonzeromultiplicationresultcongruence pfa_offset_right_zero_extended_resultnonzeromultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_zero_extended_resultnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_zero_extended_resultnonzeromultiplicationresultcongruence))))))))))))))))))

forall p a b c. (((exists pfa_gap_zero_inverse_firstinput. pfa_gap_zero_inverse_firstinput + S (a) = (p)) /\ (((exists pfa_gap_zero_inverse_firstoutput. pfa_gap_zero_inverse_firstoutput + S (b) = (p)) /\ ((((a) = 0 /\ (b) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_zero_inverse_firstnonzeromultiplicationleft. pfa_gap_zero_inverse_firstnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_zero_inverse_firstnonzeromultiplicationright. pfa_gap_zero_inverse_firstnonzeromultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_zero_inverse_firstnonzeromultiplicationresultbound. pfa_gap_zero_inverse_firstnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_inverse_firstnonzeromultiplicationresultcongruence pfa_offset_right_zero_inverse_firstnonzeromultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_zero_inverse_firstnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_zero_inverse_firstnonzeromultiplicationresultcongruence)))))))))))))))))) -> (((exists pfa_gap_zero_inverse_secondinput. pfa_gap_zero_inverse_secondinput + S (a) = (p)) /\ (((exists pfa_gap_zero_inverse_secondoutput. pfa_gap_zero_inverse_secondoutput + S (c) = (p)) /\ ((((a) = 0 /\ (c) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_zero_inverse_secondnonzeromultiplicationleft. pfa_gap_zero_inverse_secondnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_zero_inverse_secondnonzeromultiplicationright. pfa_gap_zero_inverse_secondnonzeromultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_zero_inverse_secondnonzeromultiplicationresultbound. pfa_gap_zero_inverse_secondnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_inverse_secondnonzeromultiplicationresultcongruence pfa_offset_right_zero_inverse_secondnonzeromultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_zero_inverse_secondnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_zero_inverse_secondnonzeromultiplicationresultcongruence)))))))))))))))))) -> b = c

forall p l. (forall i. (exists pfa_gap_addchoice_domain. pfa_gap_addchoice_domain + S (i) = (l)) -> exists v. (exists pft_row_addchoice_value pft_column_addchoice_value. (((i) = pft_row_addchoice_value * (p) + pft_column_addchoice_value) /\ ((((exists pfa_gap_addchoice_valueoperationleft. pfa_gap_addchoice_valueoperationleft + S (pft_row_addchoice_value) = (p)) /\ (((exists pfa_gap_addchoice_valueoperationright. pfa_gap_addchoice_valueoperationright + S (pft_column_addchoice_value) = (p)) /\ ((((exists pfa_gap_addchoice_valueoperationresultbound. pfa_gap_addchoice_valueoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addchoice_valueoperationresultcongruence pfa_offset_right_addchoice_valueoperationresultcongruence. ((pft_row_addchoice_value) + (pft_column_addchoice_value)) + (p) * pfa_offset_left_addchoice_valueoperationresultcongruence = (v) + (p) * pfa_offset_right_addchoice_valueoperationresultcongruence))))))))))))) -> (exists b c. (forall pft_index_addchoice_result. (exists pfa_gap_addchoice_resultprefix. pfa_gap_addchoice_resultprefix + S (pft_index_addchoice_result) = (l)) -> exists pft_value_addchoice_result. (((((exists ff_h_pft_addchoice_resultpointentry. ff_h_pft_addchoice_resultpointentry + S (pft_value_addchoice_result) = S ((S (pft_index_addchoice_result)) * c)) /\ exists ff_q_pft_addchoice_resultpointentry. b = ff_q_pft_addchoice_resultpointentry * S ((S (pft_index_addchoice_result)) * c) + (pft_value_addchoice_result))) /\ ((exists pft_row_addchoice_resultpointvalue pft_column_addchoice_resultpointvalue. (((pft_index_addchoice_result) = pft_row_addchoice_resultpointvalue * (p) + pft_column_addchoice_resultpointvalue) /\ ((((exists pfa_gap_addchoice_resultpointvalueoperationleft. pfa_gap_addchoice_resultpointvalueoperationleft + S (pft_row_addchoice_resultpointvalue) = (p)) /\ (((exists pfa_gap_addchoice_resultpointvalueoperationright. pfa_gap_addchoice_resultpointvalueoperationright + S (pft_column_addchoice_resultpointvalue) = (p)) /\ ((((exists pfa_gap_addchoice_resultpointvalueoperationresultbound. pfa_gap_addchoice_resultpointvalueoperationresultbound + S (pft_value_addchoice_result) = (p)) /\ ((exists pfa_offset_left_addchoice_resultpointvalueoperationresultcongruence pfa_offset_right_addchoice_resultpointvalueoperationresultcongruence. ((pft_row_addchoice_resultpointvalue) + (pft_column_addchoice_resultpointvalue)) + (p) * pfa_offset_left_addchoice_resultpointvalueoperationresultcongruence = (pft_value_addchoice_result) + (p) * pfa_offset_right_addchoice_resultpointvalueoperationresultcongruence)))))))))))))))))

forall p l. (forall i. (exists pfa_gap_multiplychoice_domain. pfa_gap_multiplychoice_domain + S (i) = (l)) -> exists v. (exists pft_row_multiplychoice_value pft_column_multiplychoice_value. (((i) = pft_row_multiplychoice_value * (p) + pft_column_multiplychoice_value) /\ ((((exists pfa_gap_multiplychoice_valueoperationleft. pfa_gap_multiplychoice_valueoperationleft + S (pft_row_multiplychoice_value) = (p)) /\ (((exists pfa_gap_multiplychoice_valueoperationright. pfa_gap_multiplychoice_valueoperationright + S (pft_column_multiplychoice_value) = (p)) /\ ((((exists pfa_gap_multiplychoice_valueoperationresultbound. pfa_gap_multiplychoice_valueoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplychoice_valueoperationresultcongruence pfa_offset_right_multiplychoice_valueoperationresultcongruence. ((pft_row_multiplychoice_value) * (pft_column_multiplychoice_value)) + (p) * pfa_offset_left_multiplychoice_valueoperationresultcongruence = (v) + (p) * pfa_offset_right_multiplychoice_valueoperationresultcongruence))))))))))))) -> (exists b c. (forall pft_index_multiplychoice_result. (exists pfa_gap_multiplychoice_resultprefix. pfa_gap_multiplychoice_resultprefix + S (pft_index_multiplychoice_result) = (l)) -> exists pft_value_multiplychoice_result. (((((exists ff_h_pft_multiplychoice_resultpointentry. ff_h_pft_multiplychoice_resultpointentry + S (pft_value_multiplychoice_result) = S ((S (pft_index_multiplychoice_result)) * c)) /\ exists ff_q_pft_multiplychoice_resultpointentry. b = ff_q_pft_multiplychoice_resultpointentry * S ((S (pft_index_multiplychoice_result)) * c) + (pft_value_multiplychoice_result))) /\ ((exists pft_row_multiplychoice_resultpointvalue pft_column_multiplychoice_resultpointvalue. (((pft_index_multiplychoice_result) = pft_row_multiplychoice_resultpointvalue * (p) + pft_column_multiplychoice_resultpointvalue) /\ ((((exists pfa_gap_multiplychoice_resultpointvalueoperationleft. pfa_gap_multiplychoice_resultpointvalueoperationleft + S (pft_row_multiplychoice_resultpointvalue) = (p)) /\ (((exists pfa_gap_multiplychoice_resultpointvalueoperationright. pfa_gap_multiplychoice_resultpointvalueoperationright + S (pft_column_multiplychoice_resultpointvalue) = (p)) /\ ((((exists pfa_gap_multiplychoice_resultpointvalueoperationresultbound. pfa_gap_multiplychoice_resultpointvalueoperationresultbound + S (pft_value_multiplychoice_result) = (p)) /\ ((exists pfa_offset_left_multiplychoice_resultpointvalueoperationresultcongruence pfa_offset_right_multiplychoice_resultpointvalueoperationresultcongruence. ((pft_row_multiplychoice_resultpointvalue) * (pft_column_multiplychoice_resultpointvalue)) + (p) * pfa_offset_left_multiplychoice_resultpointvalueoperationresultcongruence = (pft_value_multiplychoice_result) + (p) * pfa_offset_right_multiplychoice_resultpointvalueoperationresultcongruence)))))))))))))))))

forall p l. (forall i. (exists pfa_gap_negatechoice_domain. pfa_gap_negatechoice_domain + S (i) = (l)) -> exists v. (((exists pfa_gap_negatechoice_valueadditionleft. pfa_gap_negatechoice_valueadditionleft + S (i) = (p)) /\ (((exists pfa_gap_negatechoice_valueadditionright. pfa_gap_negatechoice_valueadditionright + S (v) = (p)) /\ ((((exists pfa_gap_negatechoice_valueadditionresultbound. pfa_gap_negatechoice_valueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negatechoice_valueadditionresultcongruence pfa_offset_right_negatechoice_valueadditionresultcongruence. ((i) + (v)) + (p) * pfa_offset_left_negatechoice_valueadditionresultcongruence = (0) + (p) * pfa_offset_right_negatechoice_valueadditionresultcongruence)))))))))) -> (exists b c. (forall pft_index_negatechoice_result. (exists pfa_gap_negatechoice_resultprefix. pfa_gap_negatechoice_resultprefix + S (pft_index_negatechoice_result) = (l)) -> exists pft_value_negatechoice_result. (((((exists ff_h_pft_negatechoice_resultpointentry. ff_h_pft_negatechoice_resultpointentry + S (pft_value_negatechoice_result) = S ((S (pft_index_negatechoice_result)) * c)) /\ exists ff_q_pft_negatechoice_resultpointentry. b = ff_q_pft_negatechoice_resultpointentry * S ((S (pft_index_negatechoice_result)) * c) + (pft_value_negatechoice_result))) /\ ((((exists pfa_gap_negatechoice_resultpointvalueadditionleft. pfa_gap_negatechoice_resultpointvalueadditionleft + S (pft_index_negatechoice_result) = (p)) /\ (((exists pfa_gap_negatechoice_resultpointvalueadditionright. pfa_gap_negatechoice_resultpointvalueadditionright + S (pft_value_negatechoice_result) = (p)) /\ ((((exists pfa_gap_negatechoice_resultpointvalueadditionresultbound. pfa_gap_negatechoice_resultpointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negatechoice_resultpointvalueadditionresultcongruence pfa_offset_right_negatechoice_resultpointvalueadditionresultcongruence. ((pft_index_negatechoice_result) + (pft_value_negatechoice_result)) + (p) * pfa_offset_left_negatechoice_resultpointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negatechoice_resultpointvalueadditionresultcongruence))))))))))))))

forall p l. (forall i. (exists pfa_gap_inversechoice_domain. pfa_gap_inversechoice_domain + S (i) = (l)) -> exists v. (((exists pfa_gap_inversechoice_valueinput. pfa_gap_inversechoice_valueinput + S (i) = (p)) /\ (((exists pfa_gap_inversechoice_valueoutput. pfa_gap_inversechoice_valueoutput + S (v) = (p)) /\ ((((i) = 0 /\ (v) = 0) \/ (((~((i) = 0)) /\ ((((exists pfa_gap_inversechoice_valuenonzeromultiplicationleft. pfa_gap_inversechoice_valuenonzeromultiplicationleft + S (i) = (p)) /\ (((exists pfa_gap_inversechoice_valuenonzeromultiplicationright. pfa_gap_inversechoice_valuenonzeromultiplicationright + S (v) = (p)) /\ ((((exists pfa_gap_inversechoice_valuenonzeromultiplicationresultbound. pfa_gap_inversechoice_valuenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inversechoice_valuenonzeromultiplicationresultcongruence pfa_offset_right_inversechoice_valuenonzeromultiplicationresultcongruence. ((i) * (v)) + (p) * pfa_offset_left_inversechoice_valuenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inversechoice_valuenonzeromultiplicationresultcongruence))))))))))))))))))) -> (exists b c. (forall pft_index_inversechoice_result. (exists pfa_gap_inversechoice_resultprefix. pfa_gap_inversechoice_resultprefix + S (pft_index_inversechoice_result) = (l)) -> exists pft_value_inversechoice_result. (((((exists ff_h_pft_inversechoice_resultpointentry. ff_h_pft_inversechoice_resultpointentry + S (pft_value_inversechoice_result) = S ((S (pft_index_inversechoice_result)) * c)) /\ exists ff_q_pft_inversechoice_resultpointentry. b = ff_q_pft_inversechoice_resultpointentry * S ((S (pft_index_inversechoice_result)) * c) + (pft_value_inversechoice_result))) /\ ((((exists pfa_gap_inversechoice_resultpointvalueinput. pfa_gap_inversechoice_resultpointvalueinput + S (pft_index_inversechoice_result) = (p)) /\ (((exists pfa_gap_inversechoice_resultpointvalueoutput. pfa_gap_inversechoice_resultpointvalueoutput + S (pft_value_inversechoice_result) = (p)) /\ ((((pft_index_inversechoice_result) = 0 /\ (pft_value_inversechoice_result) = 0) \/ (((~((pft_index_inversechoice_result) = 0)) /\ ((((exists pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationleft. pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationleft + S (pft_index_inversechoice_result) = (p)) /\ (((exists pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationright. pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationright + S (pft_value_inversechoice_result) = (p)) /\ ((((exists pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationresultbound. pfa_gap_inversechoice_resultpointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inversechoice_resultpointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inversechoice_resultpointvaluenonzeromultiplicationresultcongruence. ((pft_index_inversechoice_result) * (pft_value_inversechoice_result)) + (p) * pfa_offset_left_inversechoice_resultpointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inversechoice_resultpointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_addtable_domain pfa_factor_right_addtable_domain. (p) = pfa_factor_left_addtable_domain * pfa_factor_right_addtable_domain -> pfa_factor_left_addtable_domain = 1 \/ pfa_factor_right_addtable_domain = 1) -> exists b c. (forall pft_index_addtable_result. (exists pfa_gap_addtable_resultprefix. pfa_gap_addtable_resultprefix + S (pft_index_addtable_result) = ((p) * (p))) -> exists pft_value_addtable_result. (((((exists ff_h_pft_addtable_resultpointentry. ff_h_pft_addtable_resultpointentry + S (pft_value_addtable_result) = S ((S (pft_index_addtable_result)) * c)) /\ exists ff_q_pft_addtable_resultpointentry. b = ff_q_pft_addtable_resultpointentry * S ((S (pft_index_addtable_result)) * c) + (pft_value_addtable_result))) /\ ((exists pft_row_addtable_resultpointvalue pft_column_addtable_resultpointvalue. (((pft_index_addtable_result) = pft_row_addtable_resultpointvalue * (p) + pft_column_addtable_resultpointvalue) /\ ((((exists pfa_gap_addtable_resultpointvalueoperationleft. pfa_gap_addtable_resultpointvalueoperationleft + S (pft_row_addtable_resultpointvalue) = (p)) /\ (((exists pfa_gap_addtable_resultpointvalueoperationright. pfa_gap_addtable_resultpointvalueoperationright + S (pft_column_addtable_resultpointvalue) = (p)) /\ ((((exists pfa_gap_addtable_resultpointvalueoperationresultbound. pfa_gap_addtable_resultpointvalueoperationresultbound + S (pft_value_addtable_result) = (p)) /\ ((exists pfa_offset_left_addtable_resultpointvalueoperationresultcongruence pfa_offset_right_addtable_resultpointvalueoperationresultcongruence. ((pft_row_addtable_resultpointvalue) + (pft_column_addtable_resultpointvalue)) + (p) * pfa_offset_left_addtable_resultpointvalueoperationresultcongruence = (pft_value_addtable_result) + (p) * pfa_offset_right_addtable_resultpointvalueoperationresultcongruence))))))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_multiplytable_domain pfa_factor_right_multiplytable_domain. (p) = pfa_factor_left_multiplytable_domain * pfa_factor_right_multiplytable_domain -> pfa_factor_left_multiplytable_domain = 1 \/ pfa_factor_right_multiplytable_domain = 1) -> exists b c. (forall pft_index_multiplytable_result. (exists pfa_gap_multiplytable_resultprefix. pfa_gap_multiplytable_resultprefix + S (pft_index_multiplytable_result) = ((p) * (p))) -> exists pft_value_multiplytable_result. (((((exists ff_h_pft_multiplytable_resultpointentry. ff_h_pft_multiplytable_resultpointentry + S (pft_value_multiplytable_result) = S ((S (pft_index_multiplytable_result)) * c)) /\ exists ff_q_pft_multiplytable_resultpointentry. b = ff_q_pft_multiplytable_resultpointentry * S ((S (pft_index_multiplytable_result)) * c) + (pft_value_multiplytable_result))) /\ ((exists pft_row_multiplytable_resultpointvalue pft_column_multiplytable_resultpointvalue. (((pft_index_multiplytable_result) = pft_row_multiplytable_resultpointvalue * (p) + pft_column_multiplytable_resultpointvalue) /\ ((((exists pfa_gap_multiplytable_resultpointvalueoperationleft. pfa_gap_multiplytable_resultpointvalueoperationleft + S (pft_row_multiplytable_resultpointvalue) = (p)) /\ (((exists pfa_gap_multiplytable_resultpointvalueoperationright. pfa_gap_multiplytable_resultpointvalueoperationright + S (pft_column_multiplytable_resultpointvalue) = (p)) /\ ((((exists pfa_gap_multiplytable_resultpointvalueoperationresultbound. pfa_gap_multiplytable_resultpointvalueoperationresultbound + S (pft_value_multiplytable_result) = (p)) /\ ((exists pfa_offset_left_multiplytable_resultpointvalueoperationresultcongruence pfa_offset_right_multiplytable_resultpointvalueoperationresultcongruence. ((pft_row_multiplytable_resultpointvalue) * (pft_column_multiplytable_resultpointvalue)) + (p) * pfa_offset_left_multiplytable_resultpointvalueoperationresultcongruence = (pft_value_multiplytable_result) + (p) * pfa_offset_right_multiplytable_resultpointvalueoperationresultcongruence))))))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_negatetable_domain pfa_factor_right_negatetable_domain. (p) = pfa_factor_left_negatetable_domain * pfa_factor_right_negatetable_domain -> pfa_factor_left_negatetable_domain = 1 \/ pfa_factor_right_negatetable_domain = 1) -> exists b c. (forall pft_index_negatetable_result. (exists pfa_gap_negatetable_resultprefix. pfa_gap_negatetable_resultprefix + S (pft_index_negatetable_result) = (p)) -> exists pft_value_negatetable_result. (((((exists ff_h_pft_negatetable_resultpointentry. ff_h_pft_negatetable_resultpointentry + S (pft_value_negatetable_result) = S ((S (pft_index_negatetable_result)) * c)) /\ exists ff_q_pft_negatetable_resultpointentry. b = ff_q_pft_negatetable_resultpointentry * S ((S (pft_index_negatetable_result)) * c) + (pft_value_negatetable_result))) /\ ((((exists pfa_gap_negatetable_resultpointvalueadditionleft. pfa_gap_negatetable_resultpointvalueadditionleft + S (pft_index_negatetable_result) = (p)) /\ (((exists pfa_gap_negatetable_resultpointvalueadditionright. pfa_gap_negatetable_resultpointvalueadditionright + S (pft_value_negatetable_result) = (p)) /\ ((((exists pfa_gap_negatetable_resultpointvalueadditionresultbound. pfa_gap_negatetable_resultpointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negatetable_resultpointvalueadditionresultcongruence pfa_offset_right_negatetable_resultpointvalueadditionresultcongruence. ((pft_index_negatetable_result) + (pft_value_negatetable_result)) + (p) * pfa_offset_left_negatetable_resultpointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negatetable_resultpointvalueadditionresultcongruence)))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_inversetable_domain pfa_factor_right_inversetable_domain. (p) = pfa_factor_left_inversetable_domain * pfa_factor_right_inversetable_domain -> pfa_factor_left_inversetable_domain = 1 \/ pfa_factor_right_inversetable_domain = 1) -> exists b c. (forall pft_index_inversetable_result. (exists pfa_gap_inversetable_resultprefix. pfa_gap_inversetable_resultprefix + S (pft_index_inversetable_result) = (p)) -> exists pft_value_inversetable_result. (((((exists ff_h_pft_inversetable_resultpointentry. ff_h_pft_inversetable_resultpointentry + S (pft_value_inversetable_result) = S ((S (pft_index_inversetable_result)) * c)) /\ exists ff_q_pft_inversetable_resultpointentry. b = ff_q_pft_inversetable_resultpointentry * S ((S (pft_index_inversetable_result)) * c) + (pft_value_inversetable_result))) /\ ((((exists pfa_gap_inversetable_resultpointvalueinput. pfa_gap_inversetable_resultpointvalueinput + S (pft_index_inversetable_result) = (p)) /\ (((exists pfa_gap_inversetable_resultpointvalueoutput. pfa_gap_inversetable_resultpointvalueoutput + S (pft_value_inversetable_result) = (p)) /\ ((((pft_index_inversetable_result) = 0 /\ (pft_value_inversetable_result) = 0) \/ (((~((pft_index_inversetable_result) = 0)) /\ ((((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationleft. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationleft + S (pft_index_inversetable_result) = (p)) /\ (((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationright. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationright + S (pft_value_inversetable_result) = (p)) /\ ((((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationresultbound. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inversetable_resultpointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inversetable_resultpointvaluenonzeromultiplicationresultcongruence. ((pft_index_inversetable_result) * (pft_value_inversetable_result)) + (p) * pfa_offset_left_inversetable_resultpointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inversetable_resultpointvaluenonzeromultiplicationresultcongruence))))))))))))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_all_tables_domain pfa_factor_right_all_tables_domain. (p) = pfa_factor_left_all_tables_domain * pfa_factor_right_all_tables_domain -> pfa_factor_left_all_tables_domain = 1 \/ pfa_factor_right_all_tables_domain = 1) -> exists ab ac mb mc nb nc ib ic. (((forall pft_index_all_tablesadd. (exists pfa_gap_all_tablesaddprefix. pfa_gap_all_tablesaddprefix + S (pft_index_all_tablesadd) = ((p) * (p))) -> exists pft_value_all_tablesadd. (((((exists ff_h_pft_all_tablesaddpointentry. ff_h_pft_all_tablesaddpointentry + S (pft_value_all_tablesadd) = S ((S (pft_index_all_tablesadd)) * ac)) /\ exists ff_q_pft_all_tablesaddpointentry. ab = ff_q_pft_all_tablesaddpointentry * S ((S (pft_index_all_tablesadd)) * ac) + (pft_value_all_tablesadd))) /\ ((exists pft_row_all_tablesaddpointvalue pft_column_all_tablesaddpointvalue. (((pft_index_all_tablesadd) = pft_row_all_tablesaddpointvalue * (p) + pft_column_all_tablesaddpointvalue) /\ ((((exists pfa_gap_all_tablesaddpointvalueoperationleft. pfa_gap_all_tablesaddpointvalueoperationleft + S (pft_row_all_tablesaddpointvalue) = (p)) /\ (((exists pfa_gap_all_tablesaddpointvalueoperationright. pfa_gap_all_tablesaddpointvalueoperationright + S (pft_column_all_tablesaddpointvalue) = (p)) /\ ((((exists pfa_gap_all_tablesaddpointvalueoperationresultbound. pfa_gap_all_tablesaddpointvalueoperationresultbound + S (pft_value_all_tablesadd) = (p)) /\ ((exists pfa_offset_left_all_tablesaddpointvalueoperationresultcongruence pfa_offset_right_all_tablesaddpointvalueoperationresultcongruence. ((pft_row_all_tablesaddpointvalue) + (pft_column_all_tablesaddpointvalue)) + (p) * pfa_offset_left_all_tablesaddpointvalueoperationresultcongruence = (pft_value_all_tablesadd) + (p) * pfa_offset_right_all_tablesaddpointvalueoperationresultcongruence)))))))))))))))) /\ (((forall pft_index_all_tablesmultiply. (exists pfa_gap_all_tablesmultiplyprefix. pfa_gap_all_tablesmultiplyprefix + S (pft_index_all_tablesmultiply) = ((p) * (p))) -> exists pft_value_all_tablesmultiply. (((((exists ff_h_pft_all_tablesmultiplypointentry. ff_h_pft_all_tablesmultiplypointentry + S (pft_value_all_tablesmultiply) = S ((S (pft_index_all_tablesmultiply)) * mc)) /\ exists ff_q_pft_all_tablesmultiplypointentry. mb = ff_q_pft_all_tablesmultiplypointentry * S ((S (pft_index_all_tablesmultiply)) * mc) + (pft_value_all_tablesmultiply))) /\ ((exists pft_row_all_tablesmultiplypointvalue pft_column_all_tablesmultiplypointvalue. (((pft_index_all_tablesmultiply) = pft_row_all_tablesmultiplypointvalue * (p) + pft_column_all_tablesmultiplypointvalue) /\ ((((exists pfa_gap_all_tablesmultiplypointvalueoperationleft. pfa_gap_all_tablesmultiplypointvalueoperationleft + S (pft_row_all_tablesmultiplypointvalue) = (p)) /\ (((exists pfa_gap_all_tablesmultiplypointvalueoperationright. pfa_gap_all_tablesmultiplypointvalueoperationright + S (pft_column_all_tablesmultiplypointvalue) = (p)) /\ ((((exists pfa_gap_all_tablesmultiplypointvalueoperationresultbound. pfa_gap_all_tablesmultiplypointvalueoperationresultbound + S (pft_value_all_tablesmultiply) = (p)) /\ ((exists pfa_offset_left_all_tablesmultiplypointvalueoperationresultcongruence pfa_offset_right_all_tablesmultiplypointvalueoperationresultcongruence. ((pft_row_all_tablesmultiplypointvalue) * (pft_column_all_tablesmultiplypointvalue)) + (p) * pfa_offset_left_all_tablesmultiplypointvalueoperationresultcongruence = (pft_value_all_tablesmultiply) + (p) * pfa_offset_right_all_tablesmultiplypointvalueoperationresultcongruence)))))))))))))))) /\ (((forall pft_index_all_tablesnegate. (exists pfa_gap_all_tablesnegateprefix. pfa_gap_all_tablesnegateprefix + S (pft_index_all_tablesnegate) = (p)) -> exists pft_value_all_tablesnegate. (((((exists ff_h_pft_all_tablesnegatepointentry. ff_h_pft_all_tablesnegatepointentry + S (pft_value_all_tablesnegate) = S ((S (pft_index_all_tablesnegate)) * nc)) /\ exists ff_q_pft_all_tablesnegatepointentry. nb = ff_q_pft_all_tablesnegatepointentry * S ((S (pft_index_all_tablesnegate)) * nc) + (pft_value_all_tablesnegate))) /\ ((((exists pfa_gap_all_tablesnegatepointvalueadditionleft. pfa_gap_all_tablesnegatepointvalueadditionleft + S (pft_index_all_tablesnegate) = (p)) /\ (((exists pfa_gap_all_tablesnegatepointvalueadditionright. pfa_gap_all_tablesnegatepointvalueadditionright + S (pft_value_all_tablesnegate) = (p)) /\ ((((exists pfa_gap_all_tablesnegatepointvalueadditionresultbound. pfa_gap_all_tablesnegatepointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_all_tablesnegatepointvalueadditionresultcongruence pfa_offset_right_all_tablesnegatepointvalueadditionresultcongruence. ((pft_index_all_tablesnegate) + (pft_value_all_tablesnegate)) + (p) * pfa_offset_left_all_tablesnegatepointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_all_tablesnegatepointvalueadditionresultcongruence))))))))))))) /\ ((forall pft_index_all_tablesinverse. (exists pfa_gap_all_tablesinverseprefix. pfa_gap_all_tablesinverseprefix + S (pft_index_all_tablesinverse) = (p)) -> exists pft_value_all_tablesinverse. (((((exists ff_h_pft_all_tablesinversepointentry. ff_h_pft_all_tablesinversepointentry + S (pft_value_all_tablesinverse) = S ((S (pft_index_all_tablesinverse)) * ic)) /\ exists ff_q_pft_all_tablesinversepointentry. ib = ff_q_pft_all_tablesinversepointentry * S ((S (pft_index_all_tablesinverse)) * ic) + (pft_value_all_tablesinverse))) /\ ((((exists pfa_gap_all_tablesinversepointvalueinput. pfa_gap_all_tablesinversepointvalueinput + S (pft_index_all_tablesinverse) = (p)) /\ (((exists pfa_gap_all_tablesinversepointvalueoutput. pfa_gap_all_tablesinversepointvalueoutput + S (pft_value_all_tablesinverse) = (p)) /\ ((((pft_index_all_tablesinverse) = 0 /\ (pft_value_all_tablesinverse) = 0) \/ (((~((pft_index_all_tablesinverse) = 0)) /\ ((((exists pfa_gap_all_tablesinversepointvaluenonzeromultiplicationleft. pfa_gap_all_tablesinversepointvaluenonzeromultiplicationleft + S (pft_index_all_tablesinverse) = (p)) /\ (((exists pfa_gap_all_tablesinversepointvaluenonzeromultiplicationright. pfa_gap_all_tablesinversepointvaluenonzeromultiplicationright + S (pft_value_all_tablesinverse) = (p)) /\ ((((exists pfa_gap_all_tablesinversepointvaluenonzeromultiplicationresultbound. pfa_gap_all_tablesinversepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_all_tablesinversepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_all_tablesinversepointvaluenonzeromultiplicationresultcongruence. ((pft_index_all_tablesinverse) * (pft_value_all_tablesinverse)) + (p) * pfa_offset_left_all_tablesinversepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_all_tablesinversepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))))))))))

forall p a b v. (exists pfa_gap_addlookup_column. pfa_gap_addlookup_column + S (b) = (p)) -> (exists pft_row_addlookup_grid pft_column_addlookup_grid. (((a*p+b) = pft_row_addlookup_grid * (p) + pft_column_addlookup_grid) /\ ((((exists pfa_gap_addlookup_gridoperationleft. pfa_gap_addlookup_gridoperationleft + S (pft_row_addlookup_grid) = (p)) /\ (((exists pfa_gap_addlookup_gridoperationright. pfa_gap_addlookup_gridoperationright + S (pft_column_addlookup_grid) = (p)) /\ ((((exists pfa_gap_addlookup_gridoperationresultbound. pfa_gap_addlookup_gridoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addlookup_gridoperationresultcongruence pfa_offset_right_addlookup_gridoperationresultcongruence. ((pft_row_addlookup_grid) + (pft_column_addlookup_grid)) + (p) * pfa_offset_left_addlookup_gridoperationresultcongruence = (v) + (p) * pfa_offset_right_addlookup_gridoperationresultcongruence)))))))))))) -> (((exists pfa_gap_addlookup_resultleft. pfa_gap_addlookup_resultleft + S (a) = (p)) /\ (((exists pfa_gap_addlookup_resultright. pfa_gap_addlookup_resultright + S (b) = (p)) /\ ((((exists pfa_gap_addlookup_resultresultbound. pfa_gap_addlookup_resultresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addlookup_resultresultcongruence pfa_offset_right_addlookup_resultresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addlookup_resultresultcongruence = (v) + (p) * pfa_offset_right_addlookup_resultresultcongruence)))))))))

forall p B C a b v. (forall pft_index_addlookup_table. (exists pfa_gap_addlookup_tableprefix. pfa_gap_addlookup_tableprefix + S (pft_index_addlookup_table) = ((p) * (p))) -> exists pft_value_addlookup_table. (((((exists ff_h_pft_addlookup_tablepointentry. ff_h_pft_addlookup_tablepointentry + S (pft_value_addlookup_table) = S ((S (pft_index_addlookup_table)) * C)) /\ exists ff_q_pft_addlookup_tablepointentry. B = ff_q_pft_addlookup_tablepointentry * S ((S (pft_index_addlookup_table)) * C) + (pft_value_addlookup_table))) /\ ((exists pft_row_addlookup_tablepointvalue pft_column_addlookup_tablepointvalue. (((pft_index_addlookup_table) = pft_row_addlookup_tablepointvalue * (p) + pft_column_addlookup_tablepointvalue) /\ ((((exists pfa_gap_addlookup_tablepointvalueoperationleft. pfa_gap_addlookup_tablepointvalueoperationleft + S (pft_row_addlookup_tablepointvalue) = (p)) /\ (((exists pfa_gap_addlookup_tablepointvalueoperationright. pfa_gap_addlookup_tablepointvalueoperationright + S (pft_column_addlookup_tablepointvalue) = (p)) /\ ((((exists pfa_gap_addlookup_tablepointvalueoperationresultbound. pfa_gap_addlookup_tablepointvalueoperationresultbound + S (pft_value_addlookup_table) = (p)) /\ ((exists pfa_offset_left_addlookup_tablepointvalueoperationresultcongruence pfa_offset_right_addlookup_tablepointvalueoperationresultcongruence. ((pft_row_addlookup_tablepointvalue) + (pft_column_addlookup_tablepointvalue)) + (p) * pfa_offset_left_addlookup_tablepointvalueoperationresultcongruence = (pft_value_addlookup_table) + (p) * pfa_offset_right_addlookup_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_addlookup_a. pfa_gap_addlookup_a + S (a) = (p)) -> (exists pfa_gap_addlookup_b. pfa_gap_addlookup_b + S (b) = (p)) -> (((exists ff_h_pft_addlookup_at. ff_h_pft_addlookup_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_addlookup_at. B = ff_q_pft_addlookup_at * S ((S (a*p+b)) * C) + (v))) -> (((exists pfa_gap_addlookup_graphleft. pfa_gap_addlookup_graphleft + S (a) = (p)) /\ (((exists pfa_gap_addlookup_graphright. pfa_gap_addlookup_graphright + S (b) = (p)) /\ ((((exists pfa_gap_addlookup_graphresultbound. pfa_gap_addlookup_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addlookup_graphresultcongruence pfa_offset_right_addlookup_graphresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addlookup_graphresultcongruence = (v) + (p) * pfa_offset_right_addlookup_graphresultcongruence)))))))))

forall p B C a b v. (forall pft_index_addreflect_table. (exists pfa_gap_addreflect_tableprefix. pfa_gap_addreflect_tableprefix + S (pft_index_addreflect_table) = ((p) * (p))) -> exists pft_value_addreflect_table. (((((exists ff_h_pft_addreflect_tablepointentry. ff_h_pft_addreflect_tablepointentry + S (pft_value_addreflect_table) = S ((S (pft_index_addreflect_table)) * C)) /\ exists ff_q_pft_addreflect_tablepointentry. B = ff_q_pft_addreflect_tablepointentry * S ((S (pft_index_addreflect_table)) * C) + (pft_value_addreflect_table))) /\ ((exists pft_row_addreflect_tablepointvalue pft_column_addreflect_tablepointvalue. (((pft_index_addreflect_table) = pft_row_addreflect_tablepointvalue * (p) + pft_column_addreflect_tablepointvalue) /\ ((((exists pfa_gap_addreflect_tablepointvalueoperationleft. pfa_gap_addreflect_tablepointvalueoperationleft + S (pft_row_addreflect_tablepointvalue) = (p)) /\ (((exists pfa_gap_addreflect_tablepointvalueoperationright. pfa_gap_addreflect_tablepointvalueoperationright + S (pft_column_addreflect_tablepointvalue) = (p)) /\ ((((exists pfa_gap_addreflect_tablepointvalueoperationresultbound. pfa_gap_addreflect_tablepointvalueoperationresultbound + S (pft_value_addreflect_table) = (p)) /\ ((exists pfa_offset_left_addreflect_tablepointvalueoperationresultcongruence pfa_offset_right_addreflect_tablepointvalueoperationresultcongruence. ((pft_row_addreflect_tablepointvalue) + (pft_column_addreflect_tablepointvalue)) + (p) * pfa_offset_left_addreflect_tablepointvalueoperationresultcongruence = (pft_value_addreflect_table) + (p) * pfa_offset_right_addreflect_tablepointvalueoperationresultcongruence)))))))))))))))) -> (((exists pfa_gap_addreflect_graphleft. pfa_gap_addreflect_graphleft + S (a) = (p)) /\ (((exists pfa_gap_addreflect_graphright. pfa_gap_addreflect_graphright + S (b) = (p)) /\ ((((exists pfa_gap_addreflect_graphresultbound. pfa_gap_addreflect_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addreflect_graphresultcongruence pfa_offset_right_addreflect_graphresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addreflect_graphresultcongruence = (v) + (p) * pfa_offset_right_addreflect_graphresultcongruence))))))))) -> (((exists ff_h_pft_addreflect_at. ff_h_pft_addreflect_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_addreflect_at. B = ff_q_pft_addreflect_at * S ((S (a*p+b)) * C) + (v)))

forall p a b v. (exists pfa_gap_multiplylookup_column. pfa_gap_multiplylookup_column + S (b) = (p)) -> (exists pft_row_multiplylookup_grid pft_column_multiplylookup_grid. (((a*p+b) = pft_row_multiplylookup_grid * (p) + pft_column_multiplylookup_grid) /\ ((((exists pfa_gap_multiplylookup_gridoperationleft. pfa_gap_multiplylookup_gridoperationleft + S (pft_row_multiplylookup_grid) = (p)) /\ (((exists pfa_gap_multiplylookup_gridoperationright. pfa_gap_multiplylookup_gridoperationright + S (pft_column_multiplylookup_grid) = (p)) /\ ((((exists pfa_gap_multiplylookup_gridoperationresultbound. pfa_gap_multiplylookup_gridoperationresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplylookup_gridoperationresultcongruence pfa_offset_right_multiplylookup_gridoperationresultcongruence. ((pft_row_multiplylookup_grid) * (pft_column_multiplylookup_grid)) + (p) * pfa_offset_left_multiplylookup_gridoperationresultcongruence = (v) + (p) * pfa_offset_right_multiplylookup_gridoperationresultcongruence)))))))))))) -> (((exists pfa_gap_multiplylookup_resultleft. pfa_gap_multiplylookup_resultleft + S (a) = (p)) /\ (((exists pfa_gap_multiplylookup_resultright. pfa_gap_multiplylookup_resultright + S (b) = (p)) /\ ((((exists pfa_gap_multiplylookup_resultresultbound. pfa_gap_multiplylookup_resultresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplylookup_resultresultcongruence pfa_offset_right_multiplylookup_resultresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplylookup_resultresultcongruence = (v) + (p) * pfa_offset_right_multiplylookup_resultresultcongruence)))))))))

forall p B C a b v. (forall pft_index_multiplylookup_table. (exists pfa_gap_multiplylookup_tableprefix. pfa_gap_multiplylookup_tableprefix + S (pft_index_multiplylookup_table) = ((p) * (p))) -> exists pft_value_multiplylookup_table. (((((exists ff_h_pft_multiplylookup_tablepointentry. ff_h_pft_multiplylookup_tablepointentry + S (pft_value_multiplylookup_table) = S ((S (pft_index_multiplylookup_table)) * C)) /\ exists ff_q_pft_multiplylookup_tablepointentry. B = ff_q_pft_multiplylookup_tablepointentry * S ((S (pft_index_multiplylookup_table)) * C) + (pft_value_multiplylookup_table))) /\ ((exists pft_row_multiplylookup_tablepointvalue pft_column_multiplylookup_tablepointvalue. (((pft_index_multiplylookup_table) = pft_row_multiplylookup_tablepointvalue * (p) + pft_column_multiplylookup_tablepointvalue) /\ ((((exists pfa_gap_multiplylookup_tablepointvalueoperationleft. pfa_gap_multiplylookup_tablepointvalueoperationleft + S (pft_row_multiplylookup_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplylookup_tablepointvalueoperationright. pfa_gap_multiplylookup_tablepointvalueoperationright + S (pft_column_multiplylookup_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplylookup_tablepointvalueoperationresultbound. pfa_gap_multiplylookup_tablepointvalueoperationresultbound + S (pft_value_multiplylookup_table) = (p)) /\ ((exists pfa_offset_left_multiplylookup_tablepointvalueoperationresultcongruence pfa_offset_right_multiplylookup_tablepointvalueoperationresultcongruence. ((pft_row_multiplylookup_tablepointvalue) * (pft_column_multiplylookup_tablepointvalue)) + (p) * pfa_offset_left_multiplylookup_tablepointvalueoperationresultcongruence = (pft_value_multiplylookup_table) + (p) * pfa_offset_right_multiplylookup_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_multiplylookup_a. pfa_gap_multiplylookup_a + S (a) = (p)) -> (exists pfa_gap_multiplylookup_b. pfa_gap_multiplylookup_b + S (b) = (p)) -> (((exists ff_h_pft_multiplylookup_at. ff_h_pft_multiplylookup_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplylookup_at. B = ff_q_pft_multiplylookup_at * S ((S (a*p+b)) * C) + (v))) -> (((exists pfa_gap_multiplylookup_graphleft. pfa_gap_multiplylookup_graphleft + S (a) = (p)) /\ (((exists pfa_gap_multiplylookup_graphright. pfa_gap_multiplylookup_graphright + S (b) = (p)) /\ ((((exists pfa_gap_multiplylookup_graphresultbound. pfa_gap_multiplylookup_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplylookup_graphresultcongruence pfa_offset_right_multiplylookup_graphresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplylookup_graphresultcongruence = (v) + (p) * pfa_offset_right_multiplylookup_graphresultcongruence)))))))))

forall p B C a b v. (forall pft_index_multiplyreflect_table. (exists pfa_gap_multiplyreflect_tableprefix. pfa_gap_multiplyreflect_tableprefix + S (pft_index_multiplyreflect_table) = ((p) * (p))) -> exists pft_value_multiplyreflect_table. (((((exists ff_h_pft_multiplyreflect_tablepointentry. ff_h_pft_multiplyreflect_tablepointentry + S (pft_value_multiplyreflect_table) = S ((S (pft_index_multiplyreflect_table)) * C)) /\ exists ff_q_pft_multiplyreflect_tablepointentry. B = ff_q_pft_multiplyreflect_tablepointentry * S ((S (pft_index_multiplyreflect_table)) * C) + (pft_value_multiplyreflect_table))) /\ ((exists pft_row_multiplyreflect_tablepointvalue pft_column_multiplyreflect_tablepointvalue. (((pft_index_multiplyreflect_table) = pft_row_multiplyreflect_tablepointvalue * (p) + pft_column_multiplyreflect_tablepointvalue) /\ ((((exists pfa_gap_multiplyreflect_tablepointvalueoperationleft. pfa_gap_multiplyreflect_tablepointvalueoperationleft + S (pft_row_multiplyreflect_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplyreflect_tablepointvalueoperationright. pfa_gap_multiplyreflect_tablepointvalueoperationright + S (pft_column_multiplyreflect_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplyreflect_tablepointvalueoperationresultbound. pfa_gap_multiplyreflect_tablepointvalueoperationresultbound + S (pft_value_multiplyreflect_table) = (p)) /\ ((exists pfa_offset_left_multiplyreflect_tablepointvalueoperationresultcongruence pfa_offset_right_multiplyreflect_tablepointvalueoperationresultcongruence. ((pft_row_multiplyreflect_tablepointvalue) * (pft_column_multiplyreflect_tablepointvalue)) + (p) * pfa_offset_left_multiplyreflect_tablepointvalueoperationresultcongruence = (pft_value_multiplyreflect_table) + (p) * pfa_offset_right_multiplyreflect_tablepointvalueoperationresultcongruence)))))))))))))))) -> (((exists pfa_gap_multiplyreflect_graphleft. pfa_gap_multiplyreflect_graphleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyreflect_graphright. pfa_gap_multiplyreflect_graphright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyreflect_graphresultbound. pfa_gap_multiplyreflect_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplyreflect_graphresultcongruence pfa_offset_right_multiplyreflect_graphresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyreflect_graphresultcongruence = (v) + (p) * pfa_offset_right_multiplyreflect_graphresultcongruence))))))))) -> (((exists ff_h_pft_multiplyreflect_at. ff_h_pft_multiplyreflect_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplyreflect_at. B = ff_q_pft_multiplyreflect_at * S ((S (a*p+b)) * C) + (v)))

forall p B C a v. (forall pft_index_negateunary_lookup_table. (exists pfa_gap_negateunary_lookup_tableprefix. pfa_gap_negateunary_lookup_tableprefix + S (pft_index_negateunary_lookup_table) = (p)) -> exists pft_value_negateunary_lookup_table. (((((exists ff_h_pft_negateunary_lookup_tablepointentry. ff_h_pft_negateunary_lookup_tablepointentry + S (pft_value_negateunary_lookup_table) = S ((S (pft_index_negateunary_lookup_table)) * C)) /\ exists ff_q_pft_negateunary_lookup_tablepointentry. B = ff_q_pft_negateunary_lookup_tablepointentry * S ((S (pft_index_negateunary_lookup_table)) * C) + (pft_value_negateunary_lookup_table))) /\ ((((exists pfa_gap_negateunary_lookup_tablepointvalueadditionleft. pfa_gap_negateunary_lookup_tablepointvalueadditionleft + S (pft_index_negateunary_lookup_table) = (p)) /\ (((exists pfa_gap_negateunary_lookup_tablepointvalueadditionright. pfa_gap_negateunary_lookup_tablepointvalueadditionright + S (pft_value_negateunary_lookup_table) = (p)) /\ ((((exists pfa_gap_negateunary_lookup_tablepointvalueadditionresultbound. pfa_gap_negateunary_lookup_tablepointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_lookup_tablepointvalueadditionresultcongruence pfa_offset_right_negateunary_lookup_tablepointvalueadditionresultcongruence. ((pft_index_negateunary_lookup_table) + (pft_value_negateunary_lookup_table)) + (p) * pfa_offset_left_negateunary_lookup_tablepointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_lookup_tablepointvalueadditionresultcongruence))))))))))))) -> (exists pfa_gap_negateunary_lookup_bound. pfa_gap_negateunary_lookup_bound + S (a) = (p)) -> (((exists ff_h_pft_negateunary_lookup_at. ff_h_pft_negateunary_lookup_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_negateunary_lookup_at. B = ff_q_pft_negateunary_lookup_at * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_negateunary_lookup_graphadditionleft. pfa_gap_negateunary_lookup_graphadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negateunary_lookup_graphadditionright. pfa_gap_negateunary_lookup_graphadditionright + S (v) = (p)) /\ ((((exists pfa_gap_negateunary_lookup_graphadditionresultbound. pfa_gap_negateunary_lookup_graphadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_lookup_graphadditionresultcongruence pfa_offset_right_negateunary_lookup_graphadditionresultcongruence. ((a) + (v)) + (p) * pfa_offset_left_negateunary_lookup_graphadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_lookup_graphadditionresultcongruence)))))))))

forall p B C a v. (forall pft_index_negateunary_reflect_table. (exists pfa_gap_negateunary_reflect_tableprefix. pfa_gap_negateunary_reflect_tableprefix + S (pft_index_negateunary_reflect_table) = (p)) -> exists pft_value_negateunary_reflect_table. (((((exists ff_h_pft_negateunary_reflect_tablepointentry. ff_h_pft_negateunary_reflect_tablepointentry + S (pft_value_negateunary_reflect_table) = S ((S (pft_index_negateunary_reflect_table)) * C)) /\ exists ff_q_pft_negateunary_reflect_tablepointentry. B = ff_q_pft_negateunary_reflect_tablepointentry * S ((S (pft_index_negateunary_reflect_table)) * C) + (pft_value_negateunary_reflect_table))) /\ ((((exists pfa_gap_negateunary_reflect_tablepointvalueadditionleft. pfa_gap_negateunary_reflect_tablepointvalueadditionleft + S (pft_index_negateunary_reflect_table) = (p)) /\ (((exists pfa_gap_negateunary_reflect_tablepointvalueadditionright. pfa_gap_negateunary_reflect_tablepointvalueadditionright + S (pft_value_negateunary_reflect_table) = (p)) /\ ((((exists pfa_gap_negateunary_reflect_tablepointvalueadditionresultbound. pfa_gap_negateunary_reflect_tablepointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_reflect_tablepointvalueadditionresultcongruence pfa_offset_right_negateunary_reflect_tablepointvalueadditionresultcongruence. ((pft_index_negateunary_reflect_table) + (pft_value_negateunary_reflect_table)) + (p) * pfa_offset_left_negateunary_reflect_tablepointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_reflect_tablepointvalueadditionresultcongruence))))))))))))) -> (((exists pfa_gap_negateunary_reflect_graphadditionleft. pfa_gap_negateunary_reflect_graphadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negateunary_reflect_graphadditionright. pfa_gap_negateunary_reflect_graphadditionright + S (v) = (p)) /\ ((((exists pfa_gap_negateunary_reflect_graphadditionresultbound. pfa_gap_negateunary_reflect_graphadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_reflect_graphadditionresultcongruence pfa_offset_right_negateunary_reflect_graphadditionresultcongruence. ((a) + (v)) + (p) * pfa_offset_left_negateunary_reflect_graphadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_reflect_graphadditionresultcongruence))))))))) -> (((exists ff_h_pft_negateunary_reflect_at. ff_h_pft_negateunary_reflect_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_negateunary_reflect_at. B = ff_q_pft_negateunary_reflect_at * S ((S (a)) * C) + (v)))

forall p B C a v. (forall pft_index_inverseunary_lookup_table. (exists pfa_gap_inverseunary_lookup_tableprefix. pfa_gap_inverseunary_lookup_tableprefix + S (pft_index_inverseunary_lookup_table) = (p)) -> exists pft_value_inverseunary_lookup_table. (((((exists ff_h_pft_inverseunary_lookup_tablepointentry. ff_h_pft_inverseunary_lookup_tablepointentry + S (pft_value_inverseunary_lookup_table) = S ((S (pft_index_inverseunary_lookup_table)) * C)) /\ exists ff_q_pft_inverseunary_lookup_tablepointentry. B = ff_q_pft_inverseunary_lookup_tablepointentry * S ((S (pft_index_inverseunary_lookup_table)) * C) + (pft_value_inverseunary_lookup_table))) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvalueinput. pfa_gap_inverseunary_lookup_tablepointvalueinput + S (pft_index_inverseunary_lookup_table) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_tablepointvalueoutput. pfa_gap_inverseunary_lookup_tablepointvalueoutput + S (pft_value_inverseunary_lookup_table) = (p)) /\ ((((pft_index_inverseunary_lookup_table) = 0 /\ (pft_value_inverseunary_lookup_table) = 0) \/ (((~((pft_index_inverseunary_lookup_table) = 0)) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverseunary_lookup_table) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationright. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationright + S (pft_value_inverseunary_lookup_table) = (p)) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverseunary_lookup_table) * (pft_value_inverseunary_lookup_table)) + (p) * pfa_offset_left_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_inverseunary_lookup_bound. pfa_gap_inverseunary_lookup_bound + S (a) = (p)) -> (((exists ff_h_pft_inverseunary_lookup_at. ff_h_pft_inverseunary_lookup_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_inverseunary_lookup_at. B = ff_q_pft_inverseunary_lookup_at * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_inverseunary_lookup_graphinput. pfa_gap_inverseunary_lookup_graphinput + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_graphoutput. pfa_gap_inverseunary_lookup_graphoutput + S (v) = (p)) /\ ((((a) = 0 /\ (v) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationleft. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationright. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationright + S (v) = (p)) /\ ((((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationresultbound. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_lookup_graphnonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_lookup_graphnonzeromultiplicationresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverseunary_lookup_graphnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_lookup_graphnonzeromultiplicationresultcongruence))))))))))))))))))

forall p B C a v. (forall pft_index_inverseunary_reflect_table. (exists pfa_gap_inverseunary_reflect_tableprefix. pfa_gap_inverseunary_reflect_tableprefix + S (pft_index_inverseunary_reflect_table) = (p)) -> exists pft_value_inverseunary_reflect_table. (((((exists ff_h_pft_inverseunary_reflect_tablepointentry. ff_h_pft_inverseunary_reflect_tablepointentry + S (pft_value_inverseunary_reflect_table) = S ((S (pft_index_inverseunary_reflect_table)) * C)) /\ exists ff_q_pft_inverseunary_reflect_tablepointentry. B = ff_q_pft_inverseunary_reflect_tablepointentry * S ((S (pft_index_inverseunary_reflect_table)) * C) + (pft_value_inverseunary_reflect_table))) /\ ((((exists pfa_gap_inverseunary_reflect_tablepointvalueinput. pfa_gap_inverseunary_reflect_tablepointvalueinput + S (pft_index_inverseunary_reflect_table) = (p)) /\ (((exists pfa_gap_inverseunary_reflect_tablepointvalueoutput. pfa_gap_inverseunary_reflect_tablepointvalueoutput + S (pft_value_inverseunary_reflect_table) = (p)) /\ ((((pft_index_inverseunary_reflect_table) = 0 /\ (pft_value_inverseunary_reflect_table) = 0) \/ (((~((pft_index_inverseunary_reflect_table) = 0)) /\ ((((exists pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverseunary_reflect_table) = (p)) /\ (((exists pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationright. pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationright + S (pft_value_inverseunary_reflect_table) = (p)) /\ ((((exists pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverseunary_reflect_table) * (pft_value_inverseunary_reflect_table)) + (p) * pfa_offset_left_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_reflect_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (((exists pfa_gap_inverseunary_reflect_graphinput. pfa_gap_inverseunary_reflect_graphinput + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_reflect_graphoutput. pfa_gap_inverseunary_reflect_graphoutput + S (v) = (p)) /\ ((((a) = 0 /\ (v) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_inverseunary_reflect_graphnonzeromultiplicationleft. pfa_gap_inverseunary_reflect_graphnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_reflect_graphnonzeromultiplicationright. pfa_gap_inverseunary_reflect_graphnonzeromultiplicationright + S (v) = (p)) /\ ((((exists pfa_gap_inverseunary_reflect_graphnonzeromultiplicationresultbound. pfa_gap_inverseunary_reflect_graphnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_reflect_graphnonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_reflect_graphnonzeromultiplicationresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverseunary_reflect_graphnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_reflect_graphnonzeromultiplicationresultcongruence)))))))))))))))))) -> (((exists ff_h_pft_inverseunary_reflect_at. ff_h_pft_inverseunary_reflect_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_inverseunary_reflect_at. B = ff_q_pft_inverseunary_reflect_at * S ((S (a)) * C) + (v)))

forall p B C a b v. (forall pft_index_addcomm_table. (exists pfa_gap_addcomm_tableprefix. pfa_gap_addcomm_tableprefix + S (pft_index_addcomm_table) = ((p) * (p))) -> exists pft_value_addcomm_table. (((((exists ff_h_pft_addcomm_tablepointentry. ff_h_pft_addcomm_tablepointentry + S (pft_value_addcomm_table) = S ((S (pft_index_addcomm_table)) * C)) /\ exists ff_q_pft_addcomm_tablepointentry. B = ff_q_pft_addcomm_tablepointentry * S ((S (pft_index_addcomm_table)) * C) + (pft_value_addcomm_table))) /\ ((exists pft_row_addcomm_tablepointvalue pft_column_addcomm_tablepointvalue. (((pft_index_addcomm_table) = pft_row_addcomm_tablepointvalue * (p) + pft_column_addcomm_tablepointvalue) /\ ((((exists pfa_gap_addcomm_tablepointvalueoperationleft. pfa_gap_addcomm_tablepointvalueoperationleft + S (pft_row_addcomm_tablepointvalue) = (p)) /\ (((exists pfa_gap_addcomm_tablepointvalueoperationright. pfa_gap_addcomm_tablepointvalueoperationright + S (pft_column_addcomm_tablepointvalue) = (p)) /\ ((((exists pfa_gap_addcomm_tablepointvalueoperationresultbound. pfa_gap_addcomm_tablepointvalueoperationresultbound + S (pft_value_addcomm_table) = (p)) /\ ((exists pfa_offset_left_addcomm_tablepointvalueoperationresultcongruence pfa_offset_right_addcomm_tablepointvalueoperationresultcongruence. ((pft_row_addcomm_tablepointvalue) + (pft_column_addcomm_tablepointvalue)) + (p) * pfa_offset_left_addcomm_tablepointvalueoperationresultcongruence = (pft_value_addcomm_table) + (p) * pfa_offset_right_addcomm_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_addcomm_a. pfa_gap_addcomm_a + S (a) = (p)) -> (exists pfa_gap_addcomm_b. pfa_gap_addcomm_b + S (b) = (p)) -> (((exists ff_h_pft_addcomm_source. ff_h_pft_addcomm_source + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_addcomm_source. B = ff_q_pft_addcomm_source * S ((S (a*p+b)) * C) + (v))) -> (((exists ff_h_pft_addcomm_target. ff_h_pft_addcomm_target + S (v) = S ((S (b*p+a)) * C)) /\ exists ff_q_pft_addcomm_target. B = ff_q_pft_addcomm_target * S ((S (b*p+a)) * C) + (v)))

forall p B C a b c x y u v. (forall pft_index_addassoc_table. (exists pfa_gap_addassoc_tableprefix. pfa_gap_addassoc_tableprefix + S (pft_index_addassoc_table) = ((p) * (p))) -> exists pft_value_addassoc_table. (((((exists ff_h_pft_addassoc_tablepointentry. ff_h_pft_addassoc_tablepointentry + S (pft_value_addassoc_table) = S ((S (pft_index_addassoc_table)) * C)) /\ exists ff_q_pft_addassoc_tablepointentry. B = ff_q_pft_addassoc_tablepointentry * S ((S (pft_index_addassoc_table)) * C) + (pft_value_addassoc_table))) /\ ((exists pft_row_addassoc_tablepointvalue pft_column_addassoc_tablepointvalue. (((pft_index_addassoc_table) = pft_row_addassoc_tablepointvalue * (p) + pft_column_addassoc_tablepointvalue) /\ ((((exists pfa_gap_addassoc_tablepointvalueoperationleft. pfa_gap_addassoc_tablepointvalueoperationleft + S (pft_row_addassoc_tablepointvalue) = (p)) /\ (((exists pfa_gap_addassoc_tablepointvalueoperationright. pfa_gap_addassoc_tablepointvalueoperationright + S (pft_column_addassoc_tablepointvalue) = (p)) /\ ((((exists pfa_gap_addassoc_tablepointvalueoperationresultbound. pfa_gap_addassoc_tablepointvalueoperationresultbound + S (pft_value_addassoc_table) = (p)) /\ ((exists pfa_offset_left_addassoc_tablepointvalueoperationresultcongruence pfa_offset_right_addassoc_tablepointvalueoperationresultcongruence. ((pft_row_addassoc_tablepointvalue) + (pft_column_addassoc_tablepointvalue)) + (p) * pfa_offset_left_addassoc_tablepointvalueoperationresultcongruence = (pft_value_addassoc_table) + (p) * pfa_offset_right_addassoc_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_addassoc_a. pfa_gap_addassoc_a + S (a) = (p)) -> (exists pfa_gap_addassoc_b. pfa_gap_addassoc_b + S (b) = (p)) -> (exists pfa_gap_addassoc_c. pfa_gap_addassoc_c + S (c) = (p)) -> (((exists ff_h_pft_addtable_assoc0. ff_h_pft_addtable_assoc0 + S (x) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_addtable_assoc0. B = ff_q_pft_addtable_assoc0 * S ((S (a*p+b)) * C) + (x))) -> (((exists ff_h_pft_addtable_assoc1. ff_h_pft_addtable_assoc1 + S (u) = S ((S (x*p+c)) * C)) /\ exists ff_q_pft_addtable_assoc1. B = ff_q_pft_addtable_assoc1 * S ((S (x*p+c)) * C) + (u))) -> (((exists ff_h_pft_addtable_assoc2. ff_h_pft_addtable_assoc2 + S (y) = S ((S (b*p+c)) * C)) /\ exists ff_q_pft_addtable_assoc2. B = ff_q_pft_addtable_assoc2 * S ((S (b*p+c)) * C) + (y))) -> (((exists ff_h_pft_addtable_assoc3. ff_h_pft_addtable_assoc3 + S (v) = S ((S (a*p+y)) * C)) /\ exists ff_q_pft_addtable_assoc3. B = ff_q_pft_addtable_assoc3 * S ((S (a*p+y)) * C) + (v))) -> u = v

forall p B C a b v. (forall pft_index_multiplycomm_table. (exists pfa_gap_multiplycomm_tableprefix. pfa_gap_multiplycomm_tableprefix + S (pft_index_multiplycomm_table) = ((p) * (p))) -> exists pft_value_multiplycomm_table. (((((exists ff_h_pft_multiplycomm_tablepointentry. ff_h_pft_multiplycomm_tablepointentry + S (pft_value_multiplycomm_table) = S ((S (pft_index_multiplycomm_table)) * C)) /\ exists ff_q_pft_multiplycomm_tablepointentry. B = ff_q_pft_multiplycomm_tablepointentry * S ((S (pft_index_multiplycomm_table)) * C) + (pft_value_multiplycomm_table))) /\ ((exists pft_row_multiplycomm_tablepointvalue pft_column_multiplycomm_tablepointvalue. (((pft_index_multiplycomm_table) = pft_row_multiplycomm_tablepointvalue * (p) + pft_column_multiplycomm_tablepointvalue) /\ ((((exists pfa_gap_multiplycomm_tablepointvalueoperationleft. pfa_gap_multiplycomm_tablepointvalueoperationleft + S (pft_row_multiplycomm_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplycomm_tablepointvalueoperationright. pfa_gap_multiplycomm_tablepointvalueoperationright + S (pft_column_multiplycomm_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplycomm_tablepointvalueoperationresultbound. pfa_gap_multiplycomm_tablepointvalueoperationresultbound + S (pft_value_multiplycomm_table) = (p)) /\ ((exists pfa_offset_left_multiplycomm_tablepointvalueoperationresultcongruence pfa_offset_right_multiplycomm_tablepointvalueoperationresultcongruence. ((pft_row_multiplycomm_tablepointvalue) * (pft_column_multiplycomm_tablepointvalue)) + (p) * pfa_offset_left_multiplycomm_tablepointvalueoperationresultcongruence = (pft_value_multiplycomm_table) + (p) * pfa_offset_right_multiplycomm_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_multiplycomm_a. pfa_gap_multiplycomm_a + S (a) = (p)) -> (exists pfa_gap_multiplycomm_b. pfa_gap_multiplycomm_b + S (b) = (p)) -> (((exists ff_h_pft_multiplycomm_source. ff_h_pft_multiplycomm_source + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplycomm_source. B = ff_q_pft_multiplycomm_source * S ((S (a*p+b)) * C) + (v))) -> (((exists ff_h_pft_multiplycomm_target. ff_h_pft_multiplycomm_target + S (v) = S ((S (b*p+a)) * C)) /\ exists ff_q_pft_multiplycomm_target. B = ff_q_pft_multiplycomm_target * S ((S (b*p+a)) * C) + (v)))

forall p B C a b c x y u v. (forall pft_index_multiplyassoc_table. (exists pfa_gap_multiplyassoc_tableprefix. pfa_gap_multiplyassoc_tableprefix + S (pft_index_multiplyassoc_table) = ((p) * (p))) -> exists pft_value_multiplyassoc_table. (((((exists ff_h_pft_multiplyassoc_tablepointentry. ff_h_pft_multiplyassoc_tablepointentry + S (pft_value_multiplyassoc_table) = S ((S (pft_index_multiplyassoc_table)) * C)) /\ exists ff_q_pft_multiplyassoc_tablepointentry. B = ff_q_pft_multiplyassoc_tablepointentry * S ((S (pft_index_multiplyassoc_table)) * C) + (pft_value_multiplyassoc_table))) /\ ((exists pft_row_multiplyassoc_tablepointvalue pft_column_multiplyassoc_tablepointvalue. (((pft_index_multiplyassoc_table) = pft_row_multiplyassoc_tablepointvalue * (p) + pft_column_multiplyassoc_tablepointvalue) /\ ((((exists pfa_gap_multiplyassoc_tablepointvalueoperationleft. pfa_gap_multiplyassoc_tablepointvalueoperationleft + S (pft_row_multiplyassoc_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplyassoc_tablepointvalueoperationright. pfa_gap_multiplyassoc_tablepointvalueoperationright + S (pft_column_multiplyassoc_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplyassoc_tablepointvalueoperationresultbound. pfa_gap_multiplyassoc_tablepointvalueoperationresultbound + S (pft_value_multiplyassoc_table) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_tablepointvalueoperationresultcongruence pfa_offset_right_multiplyassoc_tablepointvalueoperationresultcongruence. ((pft_row_multiplyassoc_tablepointvalue) * (pft_column_multiplyassoc_tablepointvalue)) + (p) * pfa_offset_left_multiplyassoc_tablepointvalueoperationresultcongruence = (pft_value_multiplyassoc_table) + (p) * pfa_offset_right_multiplyassoc_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_multiplyassoc_a. pfa_gap_multiplyassoc_a + S (a) = (p)) -> (exists pfa_gap_multiplyassoc_b. pfa_gap_multiplyassoc_b + S (b) = (p)) -> (exists pfa_gap_multiplyassoc_c. pfa_gap_multiplyassoc_c + S (c) = (p)) -> (((exists ff_h_pft_multiplytable_assoc0. ff_h_pft_multiplytable_assoc0 + S (x) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplytable_assoc0. B = ff_q_pft_multiplytable_assoc0 * S ((S (a*p+b)) * C) + (x))) -> (((exists ff_h_pft_multiplytable_assoc1. ff_h_pft_multiplytable_assoc1 + S (u) = S ((S (x*p+c)) * C)) /\ exists ff_q_pft_multiplytable_assoc1. B = ff_q_pft_multiplytable_assoc1 * S ((S (x*p+c)) * C) + (u))) -> (((exists ff_h_pft_multiplytable_assoc2. ff_h_pft_multiplytable_assoc2 + S (y) = S ((S (b*p+c)) * C)) /\ exists ff_q_pft_multiplytable_assoc2. B = ff_q_pft_multiplytable_assoc2 * S ((S (b*p+c)) * C) + (y))) -> (((exists ff_h_pft_multiplytable_assoc3. ff_h_pft_multiplytable_assoc3 + S (v) = S ((S (a*p+y)) * C)) /\ exists ff_q_pft_multiplytable_assoc3. B = ff_q_pft_multiplytable_assoc3 * S ((S (a*p+y)) * C) + (v))) -> u = v

forall p B C. (~((p) = 1) /\ forall pfa_factor_left_inverse_zero_domain pfa_factor_right_inverse_zero_domain. (p) = pfa_factor_left_inverse_zero_domain * pfa_factor_right_inverse_zero_domain -> pfa_factor_left_inverse_zero_domain = 1 \/ pfa_factor_right_inverse_zero_domain = 1) -> (forall pft_index_inverse_zero_table. (exists pfa_gap_inverse_zero_tableprefix. pfa_gap_inverse_zero_tableprefix + S (pft_index_inverse_zero_table) = (p)) -> exists pft_value_inverse_zero_table. (((((exists ff_h_pft_inverse_zero_tablepointentry. ff_h_pft_inverse_zero_tablepointentry + S (pft_value_inverse_zero_table) = S ((S (pft_index_inverse_zero_table)) * C)) /\ exists ff_q_pft_inverse_zero_tablepointentry. B = ff_q_pft_inverse_zero_tablepointentry * S ((S (pft_index_inverse_zero_table)) * C) + (pft_value_inverse_zero_table))) /\ ((((exists pfa_gap_inverse_zero_tablepointvalueinput. pfa_gap_inverse_zero_tablepointvalueinput + S (pft_index_inverse_zero_table) = (p)) /\ (((exists pfa_gap_inverse_zero_tablepointvalueoutput. pfa_gap_inverse_zero_tablepointvalueoutput + S (pft_value_inverse_zero_table) = (p)) /\ ((((pft_index_inverse_zero_table) = 0 /\ (pft_value_inverse_zero_table) = 0) \/ (((~((pft_index_inverse_zero_table) = 0)) /\ ((((exists pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverse_zero_table) = (p)) /\ (((exists pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationright. pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationright + S (pft_value_inverse_zero_table) = (p)) /\ ((((exists pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverse_zero_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_zero_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverse_zero_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverse_zero_table) * (pft_value_inverse_zero_table)) + (p) * pfa_offset_left_inverse_zero_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_zero_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (((exists ff_h_pft_inverse_zero_entry. ff_h_pft_inverse_zero_entry + S (0) = S ((S (0)) * C)) /\ exists ff_q_pft_inverse_zero_entry. B = ff_q_pft_inverse_zero_entry * S ((S (0)) * C) + (0)))

forall p B C a v. (forall pft_index_inverse_nonzero_table. (exists pfa_gap_inverse_nonzero_tableprefix. pfa_gap_inverse_nonzero_tableprefix + S (pft_index_inverse_nonzero_table) = (p)) -> exists pft_value_inverse_nonzero_table. (((((exists ff_h_pft_inverse_nonzero_tablepointentry. ff_h_pft_inverse_nonzero_tablepointentry + S (pft_value_inverse_nonzero_table) = S ((S (pft_index_inverse_nonzero_table)) * C)) /\ exists ff_q_pft_inverse_nonzero_tablepointentry. B = ff_q_pft_inverse_nonzero_tablepointentry * S ((S (pft_index_inverse_nonzero_table)) * C) + (pft_value_inverse_nonzero_table))) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvalueinput. pfa_gap_inverse_nonzero_tablepointvalueinput + S (pft_index_inverse_nonzero_table) = (p)) /\ (((exists pfa_gap_inverse_nonzero_tablepointvalueoutput. pfa_gap_inverse_nonzero_tablepointvalueoutput + S (pft_value_inverse_nonzero_table) = (p)) /\ ((((pft_index_inverse_nonzero_table) = 0 /\ (pft_value_inverse_nonzero_table) = 0) \/ (((~((pft_index_inverse_nonzero_table) = 0)) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverse_nonzero_table) = (p)) /\ (((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationright. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationright + S (pft_value_inverse_nonzero_table) = (p)) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverse_nonzero_table) * (pft_value_inverse_nonzero_table)) + (p) * pfa_offset_left_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_inverse_nonzero_bound. pfa_gap_inverse_nonzero_bound + S (a) = (p)) -> ~(a = 0) -> (((exists ff_h_pft_inverse_nonzero_entry. ff_h_pft_inverse_nonzero_entry + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_inverse_nonzero_entry. B = ff_q_pft_inverse_nonzero_entry * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_inverse_nonzero_productleft. pfa_gap_inverse_nonzero_productleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_nonzero_productright. pfa_gap_inverse_nonzero_productright + S (v) = (p)) /\ ((((exists pfa_gap_inverse_nonzero_productresultbound. pfa_gap_inverse_nonzero_productresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_nonzero_productresultcongruence pfa_offset_right_inverse_nonzero_productresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverse_nonzero_productresultcongruence = (1) + (p) * pfa_offset_right_inverse_nonzero_productresultcongruence)))))))))

forall p A D M E a b c s x y u v. (forall pft_index_lefttable_distributive_add. (exists pfa_gap_lefttable_distributive_addprefix. pfa_gap_lefttable_distributive_addprefix + S (pft_index_lefttable_distributive_add) = ((p) * (p))) -> exists pft_value_lefttable_distributive_add. (((((exists ff_h_pft_lefttable_distributive_addpointentry. ff_h_pft_lefttable_distributive_addpointentry + S (pft_value_lefttable_distributive_add) = S ((S (pft_index_lefttable_distributive_add)) * D)) /\ exists ff_q_pft_lefttable_distributive_addpointentry. A = ff_q_pft_lefttable_distributive_addpointentry * S ((S (pft_index_lefttable_distributive_add)) * D) + (pft_value_lefttable_distributive_add))) /\ ((exists pft_row_lefttable_distributive_addpointvalue pft_column_lefttable_distributive_addpointvalue. (((pft_index_lefttable_distributive_add) = pft_row_lefttable_distributive_addpointvalue * (p) + pft_column_lefttable_distributive_addpointvalue) /\ ((((exists pfa_gap_lefttable_distributive_addpointvalueoperationleft. pfa_gap_lefttable_distributive_addpointvalueoperationleft + S (pft_row_lefttable_distributive_addpointvalue) = (p)) /\ (((exists pfa_gap_lefttable_distributive_addpointvalueoperationright. pfa_gap_lefttable_distributive_addpointvalueoperationright + S (pft_column_lefttable_distributive_addpointvalue) = (p)) /\ ((((exists pfa_gap_lefttable_distributive_addpointvalueoperationresultbound. pfa_gap_lefttable_distributive_addpointvalueoperationresultbound + S (pft_value_lefttable_distributive_add) = (p)) /\ ((exists pfa_offset_left_lefttable_distributive_addpointvalueoperationresultcongruence pfa_offset_right_lefttable_distributive_addpointvalueoperationresultcongruence. ((pft_row_lefttable_distributive_addpointvalue) + (pft_column_lefttable_distributive_addpointvalue)) + (p) * pfa_offset_left_lefttable_distributive_addpointvalueoperationresultcongruence = (pft_value_lefttable_distributive_add) + (p) * pfa_offset_right_lefttable_distributive_addpointvalueoperationresultcongruence)))))))))))))))) -> (forall pft_index_lefttable_distributive_mul. (exists pfa_gap_lefttable_distributive_mulprefix. pfa_gap_lefttable_distributive_mulprefix + S (pft_index_lefttable_distributive_mul) = ((p) * (p))) -> exists pft_value_lefttable_distributive_mul. (((((exists ff_h_pft_lefttable_distributive_mulpointentry. ff_h_pft_lefttable_distributive_mulpointentry + S (pft_value_lefttable_distributive_mul) = S ((S (pft_index_lefttable_distributive_mul)) * E)) /\ exists ff_q_pft_lefttable_distributive_mulpointentry. M = ff_q_pft_lefttable_distributive_mulpointentry * S ((S (pft_index_lefttable_distributive_mul)) * E) + (pft_value_lefttable_distributive_mul))) /\ ((exists pft_row_lefttable_distributive_mulpointvalue pft_column_lefttable_distributive_mulpointvalue. (((pft_index_lefttable_distributive_mul) = pft_row_lefttable_distributive_mulpointvalue * (p) + pft_column_lefttable_distributive_mulpointvalue) /\ ((((exists pfa_gap_lefttable_distributive_mulpointvalueoperationleft. pfa_gap_lefttable_distributive_mulpointvalueoperationleft + S (pft_row_lefttable_distributive_mulpointvalue) = (p)) /\ (((exists pfa_gap_lefttable_distributive_mulpointvalueoperationright. pfa_gap_lefttable_distributive_mulpointvalueoperationright + S (pft_column_lefttable_distributive_mulpointvalue) = (p)) /\ ((((exists pfa_gap_lefttable_distributive_mulpointvalueoperationresultbound. pfa_gap_lefttable_distributive_mulpointvalueoperationresultbound + S (pft_value_lefttable_distributive_mul) = (p)) /\ ((exists pfa_offset_left_lefttable_distributive_mulpointvalueoperationresultcongruence pfa_offset_right_lefttable_distributive_mulpointvalueoperationresultcongruence. ((pft_row_lefttable_distributive_mulpointvalue) * (pft_column_lefttable_distributive_mulpointvalue)) + (p) * pfa_offset_left_lefttable_distributive_mulpointvalueoperationresultcongruence = (pft_value_lefttable_distributive_mul) + (p) * pfa_offset_right_lefttable_distributive_mulpointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_lefttable_distributive_a. pfa_gap_lefttable_distributive_a + S (a) = (p)) -> (exists pfa_gap_lefttable_distributive_b. pfa_gap_lefttable_distributive_b + S (b) = (p)) -> (exists pfa_gap_lefttable_distributive_c. pfa_gap_lefttable_distributive_c + S (c) = (p)) -> (((exists ff_h_pft_lefttable_distributive_sum. ff_h_pft_lefttable_distributive_sum + S (s) = S ((S (b*p+c)) * D)) /\ exists ff_q_pft_lefttable_distributive_sum. A = ff_q_pft_lefttable_distributive_sum * S ((S (b*p+c)) * D) + (s))) -> (((exists ff_h_pft_lefttable_distributive_left. ff_h_pft_lefttable_distributive_left + S (u) = S ((S (a*p+s)) * E)) /\ exists ff_q_pft_lefttable_distributive_left. M = ff_q_pft_lefttable_distributive_left * S ((S (a*p+s)) * E) + (u))) -> (((exists ff_h_pft_lefttable_distributive_first. ff_h_pft_lefttable_distributive_first + S (x) = S ((S (a*p+b)) * E)) /\ exists ff_q_pft_lefttable_distributive_first. M = ff_q_pft_lefttable_distributive_first * S ((S (a*p+b)) * E) + (x))) -> (((exists ff_h_pft_lefttable_distributive_second. ff_h_pft_lefttable_distributive_second + S (y) = S ((S (a*p+c)) * E)) /\ exists ff_q_pft_lefttable_distributive_second. M = ff_q_pft_lefttable_distributive_second * S ((S (a*p+c)) * E) + (y))) -> (((exists ff_h_pft_lefttable_distributive_right. ff_h_pft_lefttable_distributive_right + S (v) = S ((S (x*p+y)) * D)) /\ exists ff_q_pft_lefttable_distributive_right. A = ff_q_pft_lefttable_distributive_right * S ((S (x*p+y)) * D) + (v))) -> u = v

forall p A D M E a b c s x y u v. (forall pft_index_righttable_distributive_add. (exists pfa_gap_righttable_distributive_addprefix. pfa_gap_righttable_distributive_addprefix + S (pft_index_righttable_distributive_add) = ((p) * (p))) -> exists pft_value_righttable_distributive_add. (((((exists ff_h_pft_righttable_distributive_addpointentry. ff_h_pft_righttable_distributive_addpointentry + S (pft_value_righttable_distributive_add) = S ((S (pft_index_righttable_distributive_add)) * D)) /\ exists ff_q_pft_righttable_distributive_addpointentry. A = ff_q_pft_righttable_distributive_addpointentry * S ((S (pft_index_righttable_distributive_add)) * D) + (pft_value_righttable_distributive_add))) /\ ((exists pft_row_righttable_distributive_addpointvalue pft_column_righttable_distributive_addpointvalue. (((pft_index_righttable_distributive_add) = pft_row_righttable_distributive_addpointvalue * (p) + pft_column_righttable_distributive_addpointvalue) /\ ((((exists pfa_gap_righttable_distributive_addpointvalueoperationleft. pfa_gap_righttable_distributive_addpointvalueoperationleft + S (pft_row_righttable_distributive_addpointvalue) = (p)) /\ (((exists pfa_gap_righttable_distributive_addpointvalueoperationright. pfa_gap_righttable_distributive_addpointvalueoperationright + S (pft_column_righttable_distributive_addpointvalue) = (p)) /\ ((((exists pfa_gap_righttable_distributive_addpointvalueoperationresultbound. pfa_gap_righttable_distributive_addpointvalueoperationresultbound + S (pft_value_righttable_distributive_add) = (p)) /\ ((exists pfa_offset_left_righttable_distributive_addpointvalueoperationresultcongruence pfa_offset_right_righttable_distributive_addpointvalueoperationresultcongruence. ((pft_row_righttable_distributive_addpointvalue) + (pft_column_righttable_distributive_addpointvalue)) + (p) * pfa_offset_left_righttable_distributive_addpointvalueoperationresultcongruence = (pft_value_righttable_distributive_add) + (p) * pfa_offset_right_righttable_distributive_addpointvalueoperationresultcongruence)))))))))))))))) -> (forall pft_index_righttable_distributive_mul. (exists pfa_gap_righttable_distributive_mulprefix. pfa_gap_righttable_distributive_mulprefix + S (pft_index_righttable_distributive_mul) = ((p) * (p))) -> exists pft_value_righttable_distributive_mul. (((((exists ff_h_pft_righttable_distributive_mulpointentry. ff_h_pft_righttable_distributive_mulpointentry + S (pft_value_righttable_distributive_mul) = S ((S (pft_index_righttable_distributive_mul)) * E)) /\ exists ff_q_pft_righttable_distributive_mulpointentry. M = ff_q_pft_righttable_distributive_mulpointentry * S ((S (pft_index_righttable_distributive_mul)) * E) + (pft_value_righttable_distributive_mul))) /\ ((exists pft_row_righttable_distributive_mulpointvalue pft_column_righttable_distributive_mulpointvalue. (((pft_index_righttable_distributive_mul) = pft_row_righttable_distributive_mulpointvalue * (p) + pft_column_righttable_distributive_mulpointvalue) /\ ((((exists pfa_gap_righttable_distributive_mulpointvalueoperationleft. pfa_gap_righttable_distributive_mulpointvalueoperationleft + S (pft_row_righttable_distributive_mulpointvalue) = (p)) /\ (((exists pfa_gap_righttable_distributive_mulpointvalueoperationright. pfa_gap_righttable_distributive_mulpointvalueoperationright + S (pft_column_righttable_distributive_mulpointvalue) = (p)) /\ ((((exists pfa_gap_righttable_distributive_mulpointvalueoperationresultbound. pfa_gap_righttable_distributive_mulpointvalueoperationresultbound + S (pft_value_righttable_distributive_mul) = (p)) /\ ((exists pfa_offset_left_righttable_distributive_mulpointvalueoperationresultcongruence pfa_offset_right_righttable_distributive_mulpointvalueoperationresultcongruence. ((pft_row_righttable_distributive_mulpointvalue) * (pft_column_righttable_distributive_mulpointvalue)) + (p) * pfa_offset_left_righttable_distributive_mulpointvalueoperationresultcongruence = (pft_value_righttable_distributive_mul) + (p) * pfa_offset_right_righttable_distributive_mulpointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_righttable_distributive_a. pfa_gap_righttable_distributive_a + S (a) = (p)) -> (exists pfa_gap_righttable_distributive_b. pfa_gap_righttable_distributive_b + S (b) = (p)) -> (exists pfa_gap_righttable_distributive_c. pfa_gap_righttable_distributive_c + S (c) = (p)) -> (((exists ff_h_pft_righttable_distributive_sum. ff_h_pft_righttable_distributive_sum + S (s) = S ((S (b*p+c)) * D)) /\ exists ff_q_pft_righttable_distributive_sum. A = ff_q_pft_righttable_distributive_sum * S ((S (b*p+c)) * D) + (s))) -> (((exists ff_h_pft_righttable_distributive_left. ff_h_pft_righttable_distributive_left + S (u) = S ((S (s*p+a)) * E)) /\ exists ff_q_pft_righttable_distributive_left. M = ff_q_pft_righttable_distributive_left * S ((S (s*p+a)) * E) + (u))) -> (((exists ff_h_pft_righttable_distributive_first. ff_h_pft_righttable_distributive_first + S (x) = S ((S (b*p+a)) * E)) /\ exists ff_q_pft_righttable_distributive_first. M = ff_q_pft_righttable_distributive_first * S ((S (b*p+a)) * E) + (x))) -> (((exists ff_h_pft_righttable_distributive_second. ff_h_pft_righttable_distributive_second + S (y) = S ((S (c*p+a)) * E)) /\ exists ff_q_pft_righttable_distributive_second. M = ff_q_pft_righttable_distributive_second * S ((S (c*p+a)) * E) + (y))) -> (((exists ff_h_pft_righttable_distributive_right. ff_h_pft_righttable_distributive_right + S (v) = S ((S (x*p+y)) * D)) /\ exists ff_q_pft_righttable_distributive_right. A = ff_q_pft_righttable_distributive_right * S ((S (x*p+y)) * D) + (v))) -> u = v

forall p b c i a. (forall pff_enumeration_index_enumeration_value_source. (exists pfa_gap_enumeration_value_sourcebound. pfa_gap_enumeration_value_sourcebound + S (pff_enumeration_index_enumeration_value_source) = (p)) -> (((exists ff_h_pft_enumeration_value_sourceentry. ff_h_pft_enumeration_value_sourceentry + S (pff_enumeration_index_enumeration_value_source) = S ((S (pff_enumeration_index_enumeration_value_source)) * c)) /\ exists ff_q_pft_enumeration_value_sourceentry. b = ff_q_pft_enumeration_value_sourceentry * S ((S (pff_enumeration_index_enumeration_value_source)) * c) + (pff_enumeration_index_enumeration_value_source)))) -> (exists pfa_gap_enumeration_value_bound. pfa_gap_enumeration_value_bound + S (i) = (p)) -> (((exists ff_h_pft_enumeration_value_at. ff_h_pft_enumeration_value_at + S (a) = S ((S (i)) * c)) /\ exists ff_q_pft_enumeration_value_at. b = ff_q_pft_enumeration_value_at * S ((S (i)) * c) + (a))) -> a = i

forall p b c. (forall pff_enumeration_index_bijection_source. (exists pfa_gap_bijection_sourcebound. pfa_gap_bijection_sourcebound + S (pff_enumeration_index_bijection_source) = (p)) -> (((exists ff_h_pft_bijection_sourceentry. ff_h_pft_bijection_sourceentry + S (pff_enumeration_index_bijection_source) = S ((S (pff_enumeration_index_bijection_source)) * c)) /\ exists ff_q_pft_bijection_sourceentry. b = ff_q_pft_bijection_sourceentry * S ((S (pff_enumeration_index_bijection_source)) * c) + (pff_enumeration_index_bijection_source)))) -> (((forall pff_enumeration_index_bijection_resultenumeration. (exists pfa_gap_bijection_resultenumerationbound. pfa_gap_bijection_resultenumerationbound + S (pff_enumeration_index_bijection_resultenumeration) = (p)) -> (((exists ff_h_pft_bijection_resultenumerationentry. ff_h_pft_bijection_resultenumerationentry + S (pff_enumeration_index_bijection_resultenumeration) = S ((S (pff_enumeration_index_bijection_resultenumeration)) * c)) /\ exists ff_q_pft_bijection_resultenumerationentry. b = ff_q_pft_bijection_resultenumerationentry * S ((S (pff_enumeration_index_bijection_resultenumeration)) * c) + (pff_enumeration_index_bijection_resultenumeration)))) /\ (((forall pff_cardinality_i_bijection_result pff_cardinality_a_bijection_result. (exists pfa_gap_bijection_resultbounded_index. pfa_gap_bijection_resultbounded_index + S (pff_cardinality_i_bijection_result) = (p)) -> (((exists ff_h_pft_bijection_resultbounded_entry. ff_h_pft_bijection_resultbounded_entry + S (pff_cardinality_a_bijection_result) = S ((S (pff_cardinality_i_bijection_result)) * c)) /\ exists ff_q_pft_bijection_resultbounded_entry. b = ff_q_pft_bijection_resultbounded_entry * S ((S (pff_cardinality_i_bijection_result)) * c) + (pff_cardinality_a_bijection_result))) -> (exists pfa_gap_bijection_resultbounded_value. pfa_gap_bijection_resultbounded_value + S (pff_cardinality_a_bijection_result) = (p))) /\ (((forall pff_cardinality_i_bijection_result pff_cardinality_j_bijection_result pff_cardinality_a_bijection_result. (exists pfa_gap_bijection_resultinjective_i. pfa_gap_bijection_resultinjective_i + S (pff_cardinality_i_bijection_result) = (p)) -> (exists pfa_gap_bijection_resultinjective_j. pfa_gap_bijection_resultinjective_j + S (pff_cardinality_j_bijection_result) = (p)) -> (((exists ff_h_pft_bijection_resultinjective_first. ff_h_pft_bijection_resultinjective_first + S (pff_cardinality_a_bijection_result) = S ((S (pff_cardinality_i_bijection_result)) * c)) /\ exists ff_q_pft_bijection_resultinjective_first. b = ff_q_pft_bijection_resultinjective_first * S ((S (pff_cardinality_i_bijection_result)) * c) + (pff_cardinality_a_bijection_result))) -> (((exists ff_h_pft_bijection_resultinjective_second. ff_h_pft_bijection_resultinjective_second + S (pff_cardinality_a_bijection_result) = S ((S (pff_cardinality_j_bijection_result)) * c)) /\ exists ff_q_pft_bijection_resultinjective_second. b = ff_q_pft_bijection_resultinjective_second * S ((S (pff_cardinality_j_bijection_result)) * c) + (pff_cardinality_a_bijection_result))) -> pff_cardinality_i_bijection_result = pff_cardinality_j_bijection_result) /\ ((forall pff_cardinality_a_bijection_result. (exists pfa_gap_bijection_resultsurjective_value. pfa_gap_bijection_resultsurjective_value + S (pff_cardinality_a_bijection_result) = (p)) -> exists pff_cardinality_i_bijection_result. (exists pfa_gap_bijection_resultsurjective_index. pfa_gap_bijection_resultsurjective_index + S (pff_cardinality_i_bijection_result) = (p)) /\ (((exists ff_h_pft_bijection_resultsurjective_entry. ff_h_pft_bijection_resultsurjective_entry + S (pff_cardinality_a_bijection_result) = S ((S (pff_cardinality_i_bijection_result)) * c)) /\ exists ff_q_pft_bijection_resultsurjective_entry. b = ff_q_pft_bijection_resultsurjective_entry * S ((S (pff_cardinality_i_bijection_result)) * c) + (pff_cardinality_a_bijection_result)))))))))))

forall p. exists b c. (((forall pff_enumeration_index_cardinality_existsenumeration. (exists pfa_gap_cardinality_existsenumerationbound. pfa_gap_cardinality_existsenumerationbound + S (pff_enumeration_index_cardinality_existsenumeration) = (p)) -> (((exists ff_h_pft_cardinality_existsenumerationentry. ff_h_pft_cardinality_existsenumerationentry + S (pff_enumeration_index_cardinality_existsenumeration) = S ((S (pff_enumeration_index_cardinality_existsenumeration)) * c)) /\ exists ff_q_pft_cardinality_existsenumerationentry. b = ff_q_pft_cardinality_existsenumerationentry * S ((S (pff_enumeration_index_cardinality_existsenumeration)) * c) + (pff_enumeration_index_cardinality_existsenumeration)))) /\ (((forall pff_cardinality_i_cardinality_exists pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existsbounded_index. pfa_gap_cardinality_existsbounded_index + S (pff_cardinality_i_cardinality_exists) = (p)) -> (((exists ff_h_pft_cardinality_existsbounded_entry. ff_h_pft_cardinality_existsbounded_entry + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsbounded_entry. b = ff_q_pft_cardinality_existsbounded_entry * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> (exists pfa_gap_cardinality_existsbounded_value. pfa_gap_cardinality_existsbounded_value + S (pff_cardinality_a_cardinality_exists) = (p))) /\ (((forall pff_cardinality_i_cardinality_exists pff_cardinality_j_cardinality_exists pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existsinjective_i. pfa_gap_cardinality_existsinjective_i + S (pff_cardinality_i_cardinality_exists) = (p)) -> (exists pfa_gap_cardinality_existsinjective_j. pfa_gap_cardinality_existsinjective_j + S (pff_cardinality_j_cardinality_exists) = (p)) -> (((exists ff_h_pft_cardinality_existsinjective_first. ff_h_pft_cardinality_existsinjective_first + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsinjective_first. b = ff_q_pft_cardinality_existsinjective_first * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> (((exists ff_h_pft_cardinality_existsinjective_second. ff_h_pft_cardinality_existsinjective_second + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_j_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsinjective_second. b = ff_q_pft_cardinality_existsinjective_second * S ((S (pff_cardinality_j_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> pff_cardinality_i_cardinality_exists = pff_cardinality_j_cardinality_exists) /\ ((forall pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existssurjective_value. pfa_gap_cardinality_existssurjective_value + S (pff_cardinality_a_cardinality_exists) = (p)) -> exists pff_cardinality_i_cardinality_exists. (exists pfa_gap_cardinality_existssurjective_index. pfa_gap_cardinality_existssurjective_index + S (pff_cardinality_i_cardinality_exists) = (p)) /\ (((exists ff_h_pft_cardinality_existssurjective_entry. ff_h_pft_cardinality_existssurjective_entry + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existssurjective_entry. b = ff_q_pft_cardinality_existssurjective_entry * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists)))))))))))

forall p b c B C n r. (((((exists ff_h_pft_trace_recode_sourcestart. ff_h_pft_trace_recode_sourcestart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_recode_sourcestart. b = ff_q_pft_trace_recode_sourcestart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_recode_sourceterminal. ff_h_pft_trace_recode_sourceterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_recode_sourceterminal. b = ff_q_pft_trace_recode_sourceterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_recode_sourcesteps. (exists pfa_gap_trace_recode_sourcestepsindex. pfa_gap_trace_recode_sourcestepsindex + S (pff_trace_index_trace_recode_sourcesteps) = (n)) -> exists pff_trace_before_trace_recode_sourcesteps pff_trace_after_trace_recode_sourcesteps. ((((exists ff_h_pft_trace_recode_sourcestepsbefore. ff_h_pft_trace_recode_sourcestepsbefore + S (pff_trace_before_trace_recode_sourcesteps) = S ((S (pff_trace_index_trace_recode_sourcesteps)) * c)) /\ exists ff_q_pft_trace_recode_sourcestepsbefore. b = ff_q_pft_trace_recode_sourcestepsbefore * S ((S (pff_trace_index_trace_recode_sourcesteps)) * c) + (pff_trace_before_trace_recode_sourcesteps))) /\ (((((exists ff_h_pft_trace_recode_sourcestepsafter. ff_h_pft_trace_recode_sourcestepsafter + S (pff_trace_after_trace_recode_sourcesteps) = S ((S (S (pff_trace_index_trace_recode_sourcesteps))) * c)) /\ exists ff_q_pft_trace_recode_sourcestepsafter. b = ff_q_pft_trace_recode_sourcestepsafter * S ((S (S (pff_trace_index_trace_recode_sourcesteps))) * c) + (pff_trace_after_trace_recode_sourcesteps))) /\ ((((exists pfa_gap_trace_recode_sourcestepsadditionleft. pfa_gap_trace_recode_sourcestepsadditionleft + S (pff_trace_before_trace_recode_sourcesteps) = (p)) /\ (((exists pfa_gap_trace_recode_sourcestepsadditionright. pfa_gap_trace_recode_sourcestepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_recode_sourcestepsadditionresultbound. pfa_gap_trace_recode_sourcestepsadditionresultbound + S (pff_trace_after_trace_recode_sourcesteps) = (p)) /\ ((exists pfa_offset_left_trace_recode_sourcestepsadditionresultcongruence pfa_offset_right_trace_recode_sourcestepsadditionresultcongruence. ((pff_trace_before_trace_recode_sourcesteps) + (1)) + (p) * pfa_offset_left_trace_recode_sourcestepsadditionresultcongruence = (pff_trace_after_trace_recode_sourcesteps) + (p) * pfa_offset_right_trace_recode_sourcestepsadditionresultcongruence))))))))))))))))))) -> (forall i v. (exists pfa_gap_trace_recode_bound. pfa_gap_trace_recode_bound + S (i) = (S n)) -> (((exists ff_h_pft_trace_recode_old. ff_h_pft_trace_recode_old + S (v) = S ((S (i)) * c)) /\ exists ff_q_pft_trace_recode_old. b = ff_q_pft_trace_recode_old * S ((S (i)) * c) + (v))) -> (((exists ff_h_pft_trace_recode_new. ff_h_pft_trace_recode_new + S (v) = S ((S (i)) * C)) /\ exists ff_q_pft_trace_recode_new. B = ff_q_pft_trace_recode_new * S ((S (i)) * C) + (v)))) -> (((((exists ff_h_pft_trace_recode_targetstart. ff_h_pft_trace_recode_targetstart + S (0) = S ((S (0)) * C)) /\ exists ff_q_pft_trace_recode_targetstart. B = ff_q_pft_trace_recode_targetstart * S ((S (0)) * C) + (0))) /\ (((((exists ff_h_pft_trace_recode_targetterminal. ff_h_pft_trace_recode_targetterminal + S (r) = S ((S (n)) * C)) /\ exists ff_q_pft_trace_recode_targetterminal. B = ff_q_pft_trace_recode_targetterminal * S ((S (n)) * C) + (r))) /\ ((forall pff_trace_index_trace_recode_targetsteps. (exists pfa_gap_trace_recode_targetstepsindex. pfa_gap_trace_recode_targetstepsindex + S (pff_trace_index_trace_recode_targetsteps) = (n)) -> exists pff_trace_before_trace_recode_targetsteps pff_trace_after_trace_recode_targetsteps. ((((exists ff_h_pft_trace_recode_targetstepsbefore. ff_h_pft_trace_recode_targetstepsbefore + S (pff_trace_before_trace_recode_targetsteps) = S ((S (pff_trace_index_trace_recode_targetsteps)) * C)) /\ exists ff_q_pft_trace_recode_targetstepsbefore. B = ff_q_pft_trace_recode_targetstepsbefore * S ((S (pff_trace_index_trace_recode_targetsteps)) * C) + (pff_trace_before_trace_recode_targetsteps))) /\ (((((exists ff_h_pft_trace_recode_targetstepsafter. ff_h_pft_trace_recode_targetstepsafter + S (pff_trace_after_trace_recode_targetsteps) = S ((S (S (pff_trace_index_trace_recode_targetsteps))) * C)) /\ exists ff_q_pft_trace_recode_targetstepsafter. B = ff_q_pft_trace_recode_targetstepsafter * S ((S (S (pff_trace_index_trace_recode_targetsteps))) * C) + (pff_trace_after_trace_recode_targetsteps))) /\ ((((exists pfa_gap_trace_recode_targetstepsadditionleft. pfa_gap_trace_recode_targetstepsadditionleft + S (pff_trace_before_trace_recode_targetsteps) = (p)) /\ (((exists pfa_gap_trace_recode_targetstepsadditionright. pfa_gap_trace_recode_targetstepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_recode_targetstepsadditionresultbound. pfa_gap_trace_recode_targetstepsadditionresultbound + S (pff_trace_after_trace_recode_targetsteps) = (p)) /\ ((exists pfa_offset_left_trace_recode_targetstepsadditionresultcongruence pfa_offset_right_trace_recode_targetstepsadditionresultcongruence. ((pff_trace_before_trace_recode_targetsteps) + (1)) + (p) * pfa_offset_left_trace_recode_targetstepsadditionresultcongruence = (pff_trace_after_trace_recode_targetsteps) + (p) * pfa_offset_right_trace_recode_targetstepsadditionresultcongruence)))))))))))))))))))

forall p b c n r s. (((((exists ff_h_pft_trace_successor_sourcestart. ff_h_pft_trace_successor_sourcestart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_successor_sourcestart. b = ff_q_pft_trace_successor_sourcestart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_successor_sourceterminal. ff_h_pft_trace_successor_sourceterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_successor_sourceterminal. b = ff_q_pft_trace_successor_sourceterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_successor_sourcesteps. (exists pfa_gap_trace_successor_sourcestepsindex. pfa_gap_trace_successor_sourcestepsindex + S (pff_trace_index_trace_successor_sourcesteps) = (n)) -> exists pff_trace_before_trace_successor_sourcesteps pff_trace_after_trace_successor_sourcesteps. ((((exists ff_h_pft_trace_successor_sourcestepsbefore. ff_h_pft_trace_successor_sourcestepsbefore + S (pff_trace_before_trace_successor_sourcesteps) = S ((S (pff_trace_index_trace_successor_sourcesteps)) * c)) /\ exists ff_q_pft_trace_successor_sourcestepsbefore. b = ff_q_pft_trace_successor_sourcestepsbefore * S ((S (pff_trace_index_trace_successor_sourcesteps)) * c) + (pff_trace_before_trace_successor_sourcesteps))) /\ (((((exists ff_h_pft_trace_successor_sourcestepsafter. ff_h_pft_trace_successor_sourcestepsafter + S (pff_trace_after_trace_successor_sourcesteps) = S ((S (S (pff_trace_index_trace_successor_sourcesteps))) * c)) /\ exists ff_q_pft_trace_successor_sourcestepsafter. b = ff_q_pft_trace_successor_sourcestepsafter * S ((S (S (pff_trace_index_trace_successor_sourcesteps))) * c) + (pff_trace_after_trace_successor_sourcesteps))) /\ ((((exists pfa_gap_trace_successor_sourcestepsadditionleft. pfa_gap_trace_successor_sourcestepsadditionleft + S (pff_trace_before_trace_successor_sourcesteps) = (p)) /\ (((exists pfa_gap_trace_successor_sourcestepsadditionright. pfa_gap_trace_successor_sourcestepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_successor_sourcestepsadditionresultbound. pfa_gap_trace_successor_sourcestepsadditionresultbound + S (pff_trace_after_trace_successor_sourcesteps) = (p)) /\ ((exists pfa_offset_left_trace_successor_sourcestepsadditionresultcongruence pfa_offset_right_trace_successor_sourcestepsadditionresultcongruence. ((pff_trace_before_trace_successor_sourcesteps) + (1)) + (p) * pfa_offset_left_trace_successor_sourcestepsadditionresultcongruence = (pff_trace_after_trace_successor_sourcesteps) + (p) * pfa_offset_right_trace_successor_sourcestepsadditionresultcongruence))))))))))))))))))) -> (((exists pfa_gap_trace_successor_addleft. pfa_gap_trace_successor_addleft + S (r) = (p)) /\ (((exists pfa_gap_trace_successor_addright. pfa_gap_trace_successor_addright + S (1) = (p)) /\ ((((exists pfa_gap_trace_successor_addresultbound. pfa_gap_trace_successor_addresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_trace_successor_addresultcongruence pfa_offset_right_trace_successor_addresultcongruence. ((r) + (1)) + (p) * pfa_offset_left_trace_successor_addresultcongruence = (s) + (p) * pfa_offset_right_trace_successor_addresultcongruence))))))))) -> exists B C. (((((exists ff_h_pft_trace_successor_resultstart. ff_h_pft_trace_successor_resultstart + S (0) = S ((S (0)) * C)) /\ exists ff_q_pft_trace_successor_resultstart. B = ff_q_pft_trace_successor_resultstart * S ((S (0)) * C) + (0))) /\ (((((exists ff_h_pft_trace_successor_resultterminal. ff_h_pft_trace_successor_resultterminal + S (s) = S ((S (S n)) * C)) /\ exists ff_q_pft_trace_successor_resultterminal. B = ff_q_pft_trace_successor_resultterminal * S ((S (S n)) * C) + (s))) /\ ((forall pff_trace_index_trace_successor_resultsteps. (exists pfa_gap_trace_successor_resultstepsindex. pfa_gap_trace_successor_resultstepsindex + S (pff_trace_index_trace_successor_resultsteps) = (S n)) -> exists pff_trace_before_trace_successor_resultsteps pff_trace_after_trace_successor_resultsteps. ((((exists ff_h_pft_trace_successor_resultstepsbefore. ff_h_pft_trace_successor_resultstepsbefore + S (pff_trace_before_trace_successor_resultsteps) = S ((S (pff_trace_index_trace_successor_resultsteps)) * C)) /\ exists ff_q_pft_trace_successor_resultstepsbefore. B = ff_q_pft_trace_successor_resultstepsbefore * S ((S (pff_trace_index_trace_successor_resultsteps)) * C) + (pff_trace_before_trace_successor_resultsteps))) /\ (((((exists ff_h_pft_trace_successor_resultstepsafter. ff_h_pft_trace_successor_resultstepsafter + S (pff_trace_after_trace_successor_resultsteps) = S ((S (S (pff_trace_index_trace_successor_resultsteps))) * C)) /\ exists ff_q_pft_trace_successor_resultstepsafter. B = ff_q_pft_trace_successor_resultstepsafter * S ((S (S (pff_trace_index_trace_successor_resultsteps))) * C) + (pff_trace_after_trace_successor_resultsteps))) /\ ((((exists pfa_gap_trace_successor_resultstepsadditionleft. pfa_gap_trace_successor_resultstepsadditionleft + S (pff_trace_before_trace_successor_resultsteps) = (p)) /\ (((exists pfa_gap_trace_successor_resultstepsadditionright. pfa_gap_trace_successor_resultstepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_successor_resultstepsadditionresultbound. pfa_gap_trace_successor_resultstepsadditionresultbound + S (pff_trace_after_trace_successor_resultsteps) = (p)) /\ ((exists pfa_offset_left_trace_successor_resultstepsadditionresultcongruence pfa_offset_right_trace_successor_resultstepsadditionresultcongruence. ((pff_trace_before_trace_successor_resultsteps) + (1)) + (p) * pfa_offset_left_trace_successor_resultstepsadditionresultcongruence = (pff_trace_after_trace_successor_resultsteps) + (p) * pfa_offset_right_trace_successor_resultstepsadditionresultcongruence)))))))))))))))))))

forall p n b c r. (~((p) = 1) /\ forall pfa_factor_left_trace_bridge_domain pfa_factor_right_trace_bridge_domain. (p) = pfa_factor_left_trace_bridge_domain * pfa_factor_right_trace_bridge_domain -> pfa_factor_left_trace_bridge_domain = 1 \/ pfa_factor_right_trace_bridge_domain = 1) -> (((((exists ff_h_pft_trace_bridge_sourcestart. ff_h_pft_trace_bridge_sourcestart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_bridge_sourcestart. b = ff_q_pft_trace_bridge_sourcestart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_bridge_sourceterminal. ff_h_pft_trace_bridge_sourceterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_bridge_sourceterminal. b = ff_q_pft_trace_bridge_sourceterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_bridge_sourcesteps. (exists pfa_gap_trace_bridge_sourcestepsindex. pfa_gap_trace_bridge_sourcestepsindex + S (pff_trace_index_trace_bridge_sourcesteps) = (n)) -> exists pff_trace_before_trace_bridge_sourcesteps pff_trace_after_trace_bridge_sourcesteps. ((((exists ff_h_pft_trace_bridge_sourcestepsbefore. ff_h_pft_trace_bridge_sourcestepsbefore + S (pff_trace_before_trace_bridge_sourcesteps) = S ((S (pff_trace_index_trace_bridge_sourcesteps)) * c)) /\ exists ff_q_pft_trace_bridge_sourcestepsbefore. b = ff_q_pft_trace_bridge_sourcestepsbefore * S ((S (pff_trace_index_trace_bridge_sourcesteps)) * c) + (pff_trace_before_trace_bridge_sourcesteps))) /\ (((((exists ff_h_pft_trace_bridge_sourcestepsafter. ff_h_pft_trace_bridge_sourcestepsafter + S (pff_trace_after_trace_bridge_sourcesteps) = S ((S (S (pff_trace_index_trace_bridge_sourcesteps))) * c)) /\ exists ff_q_pft_trace_bridge_sourcestepsafter. b = ff_q_pft_trace_bridge_sourcestepsafter * S ((S (S (pff_trace_index_trace_bridge_sourcesteps))) * c) + (pff_trace_after_trace_bridge_sourcesteps))) /\ ((((exists pfa_gap_trace_bridge_sourcestepsadditionleft. pfa_gap_trace_bridge_sourcestepsadditionleft + S (pff_trace_before_trace_bridge_sourcesteps) = (p)) /\ (((exists pfa_gap_trace_bridge_sourcestepsadditionright. pfa_gap_trace_bridge_sourcestepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_bridge_sourcestepsadditionresultbound. pfa_gap_trace_bridge_sourcestepsadditionresultbound + S (pff_trace_after_trace_bridge_sourcesteps) = (p)) /\ ((exists pfa_offset_left_trace_bridge_sourcestepsadditionresultcongruence pfa_offset_right_trace_bridge_sourcestepsadditionresultcongruence. ((pff_trace_before_trace_bridge_sourcesteps) + (1)) + (p) * pfa_offset_left_trace_bridge_sourcestepsadditionresultcongruence = (pff_trace_after_trace_bridge_sourcesteps) + (p) * pfa_offset_right_trace_bridge_sourcestepsadditionresultcongruence))))))))))))))))))) -> (((exists pfa_gap_trace_bridge_resultbound. pfa_gap_trace_bridge_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_trace_bridge_resultcongruence pfa_offset_right_trace_bridge_resultcongruence. (n) + (p) * pfa_offset_left_trace_bridge_resultcongruence = (r) + (p) * pfa_offset_right_trace_bridge_resultcongruence))))

forall p b c n r. (~((p) = 1) /\ forall pfa_factor_left_trace_bounded_domain pfa_factor_right_trace_bounded_domain. (p) = pfa_factor_left_trace_bounded_domain * pfa_factor_right_trace_bounded_domain -> pfa_factor_left_trace_bounded_domain = 1 \/ pfa_factor_right_trace_bounded_domain = 1) -> (((((exists ff_h_pft_trace_bounded_sourcestart. ff_h_pft_trace_bounded_sourcestart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_bounded_sourcestart. b = ff_q_pft_trace_bounded_sourcestart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_bounded_sourceterminal. ff_h_pft_trace_bounded_sourceterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_bounded_sourceterminal. b = ff_q_pft_trace_bounded_sourceterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_bounded_sourcesteps. (exists pfa_gap_trace_bounded_sourcestepsindex. pfa_gap_trace_bounded_sourcestepsindex + S (pff_trace_index_trace_bounded_sourcesteps) = (n)) -> exists pff_trace_before_trace_bounded_sourcesteps pff_trace_after_trace_bounded_sourcesteps. ((((exists ff_h_pft_trace_bounded_sourcestepsbefore. ff_h_pft_trace_bounded_sourcestepsbefore + S (pff_trace_before_trace_bounded_sourcesteps) = S ((S (pff_trace_index_trace_bounded_sourcesteps)) * c)) /\ exists ff_q_pft_trace_bounded_sourcestepsbefore. b = ff_q_pft_trace_bounded_sourcestepsbefore * S ((S (pff_trace_index_trace_bounded_sourcesteps)) * c) + (pff_trace_before_trace_bounded_sourcesteps))) /\ (((((exists ff_h_pft_trace_bounded_sourcestepsafter. ff_h_pft_trace_bounded_sourcestepsafter + S (pff_trace_after_trace_bounded_sourcesteps) = S ((S (S (pff_trace_index_trace_bounded_sourcesteps))) * c)) /\ exists ff_q_pft_trace_bounded_sourcestepsafter. b = ff_q_pft_trace_bounded_sourcestepsafter * S ((S (S (pff_trace_index_trace_bounded_sourcesteps))) * c) + (pff_trace_after_trace_bounded_sourcesteps))) /\ ((((exists pfa_gap_trace_bounded_sourcestepsadditionleft. pfa_gap_trace_bounded_sourcestepsadditionleft + S (pff_trace_before_trace_bounded_sourcesteps) = (p)) /\ (((exists pfa_gap_trace_bounded_sourcestepsadditionright. pfa_gap_trace_bounded_sourcestepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_bounded_sourcestepsadditionresultbound. pfa_gap_trace_bounded_sourcestepsadditionresultbound + S (pff_trace_after_trace_bounded_sourcesteps) = (p)) /\ ((exists pfa_offset_left_trace_bounded_sourcestepsadditionresultcongruence pfa_offset_right_trace_bounded_sourcestepsadditionresultcongruence. ((pff_trace_before_trace_bounded_sourcesteps) + (1)) + (p) * pfa_offset_left_trace_bounded_sourcestepsadditionresultcongruence = (pff_trace_after_trace_bounded_sourcesteps) + (p) * pfa_offset_right_trace_bounded_sourcestepsadditionresultcongruence))))))))))))))))))) -> (exists pfa_gap_trace_bounded_result. pfa_gap_trace_bounded_result + S (r) = (p))

forall p n. (~((p) = 1) /\ forall pfa_factor_left_trace_exists_domain pfa_factor_right_trace_exists_domain. (p) = pfa_factor_left_trace_exists_domain * pfa_factor_right_trace_exists_domain -> pfa_factor_left_trace_exists_domain = 1 \/ pfa_factor_right_trace_exists_domain = 1) -> exists b c r. (((((exists ff_h_pft_trace_exists_resultstart. ff_h_pft_trace_exists_resultstart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_exists_resultstart. b = ff_q_pft_trace_exists_resultstart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_exists_resultterminal. ff_h_pft_trace_exists_resultterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_exists_resultterminal. b = ff_q_pft_trace_exists_resultterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_exists_resultsteps. (exists pfa_gap_trace_exists_resultstepsindex. pfa_gap_trace_exists_resultstepsindex + S (pff_trace_index_trace_exists_resultsteps) = (n)) -> exists pff_trace_before_trace_exists_resultsteps pff_trace_after_trace_exists_resultsteps. ((((exists ff_h_pft_trace_exists_resultstepsbefore. ff_h_pft_trace_exists_resultstepsbefore + S (pff_trace_before_trace_exists_resultsteps) = S ((S (pff_trace_index_trace_exists_resultsteps)) * c)) /\ exists ff_q_pft_trace_exists_resultstepsbefore. b = ff_q_pft_trace_exists_resultstepsbefore * S ((S (pff_trace_index_trace_exists_resultsteps)) * c) + (pff_trace_before_trace_exists_resultsteps))) /\ (((((exists ff_h_pft_trace_exists_resultstepsafter. ff_h_pft_trace_exists_resultstepsafter + S (pff_trace_after_trace_exists_resultsteps) = S ((S (S (pff_trace_index_trace_exists_resultsteps))) * c)) /\ exists ff_q_pft_trace_exists_resultstepsafter. b = ff_q_pft_trace_exists_resultstepsafter * S ((S (S (pff_trace_index_trace_exists_resultsteps))) * c) + (pff_trace_after_trace_exists_resultsteps))) /\ ((((exists pfa_gap_trace_exists_resultstepsadditionleft. pfa_gap_trace_exists_resultstepsadditionleft + S (pff_trace_before_trace_exists_resultsteps) = (p)) /\ (((exists pfa_gap_trace_exists_resultstepsadditionright. pfa_gap_trace_exists_resultstepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_exists_resultstepsadditionresultbound. pfa_gap_trace_exists_resultstepsadditionresultbound + S (pff_trace_after_trace_exists_resultsteps) = (p)) /\ ((exists pfa_offset_left_trace_exists_resultstepsadditionresultcongruence pfa_offset_right_trace_exists_resultstepsadditionresultcongruence. ((pff_trace_before_trace_exists_resultsteps) + (1)) + (p) * pfa_offset_left_trace_exists_resultstepsadditionresultcongruence = (pff_trace_after_trace_exists_resultsteps) + (p) * pfa_offset_right_trace_exists_resultstepsadditionresultcongruence)))))))))))))))))))

forall p n r. (~((p) = 1) /\ forall pfa_factor_left_multiple_residue_domain pfa_factor_right_multiple_residue_domain. (p) = pfa_factor_left_multiple_residue_domain * pfa_factor_right_multiple_residue_domain -> pfa_factor_left_multiple_residue_domain = 1 \/ pfa_factor_right_multiple_residue_domain = 1) -> (exists pff_history_code_multiple_residue_source pff_history_scale_multiple_residue_source. (((((exists ff_h_pft_multiple_residue_sourcehistorystart. ff_h_pft_multiple_residue_sourcehistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystart. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystart * S ((S (0)) * pff_history_scale_multiple_residue_source) + (0))) /\ (((((exists ff_h_pft_multiple_residue_sourcehistoryterminal. ff_h_pft_multiple_residue_sourcehistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistoryterminal. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistoryterminal * S ((S (n)) * pff_history_scale_multiple_residue_source) + (r))) /\ ((forall pff_trace_index_multiple_residue_sourcehistorysteps. (exists pfa_gap_multiple_residue_sourcehistorystepsindex. pfa_gap_multiple_residue_sourcehistorystepsindex + S (pff_trace_index_multiple_residue_sourcehistorysteps) = (n)) -> exists pff_trace_before_multiple_residue_sourcehistorysteps pff_trace_after_multiple_residue_sourcehistorysteps. ((((exists ff_h_pft_multiple_residue_sourcehistorystepsbefore. ff_h_pft_multiple_residue_sourcehistorystepsbefore + S (pff_trace_before_multiple_residue_sourcehistorysteps) = S ((S (pff_trace_index_multiple_residue_sourcehistorysteps)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystepsbefore. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystepsbefore * S ((S (pff_trace_index_multiple_residue_sourcehistorysteps)) * pff_history_scale_multiple_residue_source) + (pff_trace_before_multiple_residue_sourcehistorysteps))) /\ (((((exists ff_h_pft_multiple_residue_sourcehistorystepsafter. ff_h_pft_multiple_residue_sourcehistorystepsafter + S (pff_trace_after_multiple_residue_sourcehistorysteps) = S ((S (S (pff_trace_index_multiple_residue_sourcehistorysteps))) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystepsafter. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystepsafter * S ((S (S (pff_trace_index_multiple_residue_sourcehistorysteps))) * pff_history_scale_multiple_residue_source) + (pff_trace_after_multiple_residue_sourcehistorysteps))) /\ ((((exists pfa_gap_multiple_residue_sourcehistorystepsadditionleft. pfa_gap_multiple_residue_sourcehistorystepsadditionleft + S (pff_trace_before_multiple_residue_sourcehistorysteps) = (p)) /\ (((exists pfa_gap_multiple_residue_sourcehistorystepsadditionright. pfa_gap_multiple_residue_sourcehistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_residue_sourcehistorystepsadditionresultbound. pfa_gap_multiple_residue_sourcehistorystepsadditionresultbound + S (pff_trace_after_multiple_residue_sourcehistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_residue_sourcehistorystepsadditionresultcongruence pfa_offset_right_multiple_residue_sourcehistorystepsadditionresultcongruence. ((pff_trace_before_multiple_residue_sourcehistorysteps) + (1)) + (p) * pfa_offset_left_multiple_residue_sourcehistorystepsadditionresultcongruence = (pff_trace_after_multiple_residue_sourcehistorysteps) + (p) * pfa_offset_right_multiple_residue_sourcehistorystepsadditionresultcongruence)))))))))))))))))))) -> (((exists pfa_gap_multiple_residue_resultbound. pfa_gap_multiple_residue_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_multiple_residue_resultcongruence pfa_offset_right_multiple_residue_resultcongruence. (n) + (p) * pfa_offset_left_multiple_residue_resultcongruence = (r) + (p) * pfa_offset_right_multiple_residue_resultcongruence))))

forall p n r. (~((p) = 1) /\ forall pfa_factor_left_multiple_construct_domain pfa_factor_right_multiple_construct_domain. (p) = pfa_factor_left_multiple_construct_domain * pfa_factor_right_multiple_construct_domain -> pfa_factor_left_multiple_construct_domain = 1 \/ pfa_factor_right_multiple_construct_domain = 1) -> (((exists pfa_gap_multiple_construct_residuebound. pfa_gap_multiple_construct_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_multiple_construct_residuecongruence pfa_offset_right_multiple_construct_residuecongruence. (n) + (p) * pfa_offset_left_multiple_construct_residuecongruence = (r) + (p) * pfa_offset_right_multiple_construct_residuecongruence)))) -> (exists pff_history_code_multiple_construct_result pff_history_scale_multiple_construct_result. (((((exists ff_h_pft_multiple_construct_resulthistorystart. ff_h_pft_multiple_construct_resulthistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_construct_result)) /\ exists ff_q_pft_multiple_construct_resulthistorystart. pff_history_code_multiple_construct_result = ff_q_pft_multiple_construct_resulthistorystart * S ((S (0)) * pff_history_scale_multiple_construct_result) + (0))) /\ (((((exists ff_h_pft_multiple_construct_resulthistoryterminal. ff_h_pft_multiple_construct_resulthistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_construct_result)) /\ exists ff_q_pft_multiple_construct_resulthistoryterminal. pff_history_code_multiple_construct_result = ff_q_pft_multiple_construct_resulthistoryterminal * S ((S (n)) * pff_history_scale_multiple_construct_result) + (r))) /\ ((forall pff_trace_index_multiple_construct_resulthistorysteps. (exists pfa_gap_multiple_construct_resulthistorystepsindex. pfa_gap_multiple_construct_resulthistorystepsindex + S (pff_trace_index_multiple_construct_resulthistorysteps) = (n)) -> exists pff_trace_before_multiple_construct_resulthistorysteps pff_trace_after_multiple_construct_resulthistorysteps. ((((exists ff_h_pft_multiple_construct_resulthistorystepsbefore. ff_h_pft_multiple_construct_resulthistorystepsbefore + S (pff_trace_before_multiple_construct_resulthistorysteps) = S ((S (pff_trace_index_multiple_construct_resulthistorysteps)) * pff_history_scale_multiple_construct_result)) /\ exists ff_q_pft_multiple_construct_resulthistorystepsbefore. pff_history_code_multiple_construct_result = ff_q_pft_multiple_construct_resulthistorystepsbefore * S ((S (pff_trace_index_multiple_construct_resulthistorysteps)) * pff_history_scale_multiple_construct_result) + (pff_trace_before_multiple_construct_resulthistorysteps))) /\ (((((exists ff_h_pft_multiple_construct_resulthistorystepsafter. ff_h_pft_multiple_construct_resulthistorystepsafter + S (pff_trace_after_multiple_construct_resulthistorysteps) = S ((S (S (pff_trace_index_multiple_construct_resulthistorysteps))) * pff_history_scale_multiple_construct_result)) /\ exists ff_q_pft_multiple_construct_resulthistorystepsafter. pff_history_code_multiple_construct_result = ff_q_pft_multiple_construct_resulthistorystepsafter * S ((S (S (pff_trace_index_multiple_construct_resulthistorysteps))) * pff_history_scale_multiple_construct_result) + (pff_trace_after_multiple_construct_resulthistorysteps))) /\ ((((exists pfa_gap_multiple_construct_resulthistorystepsadditionleft. pfa_gap_multiple_construct_resulthistorystepsadditionleft + S (pff_trace_before_multiple_construct_resulthistorysteps) = (p)) /\ (((exists pfa_gap_multiple_construct_resulthistorystepsadditionright. pfa_gap_multiple_construct_resulthistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_construct_resulthistorystepsadditionresultbound. pfa_gap_multiple_construct_resulthistorystepsadditionresultbound + S (pff_trace_after_multiple_construct_resulthistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_construct_resulthistorystepsadditionresultcongruence pfa_offset_right_multiple_construct_resulthistorystepsadditionresultcongruence. ((pff_trace_before_multiple_construct_resulthistorysteps) + (1)) + (p) * pfa_offset_left_multiple_construct_resulthistorystepsadditionresultcongruence = (pff_trace_after_multiple_construct_resulthistorysteps) + (p) * pfa_offset_right_multiple_construct_resulthistorystepsadditionresultcongruence))))))))))))))))))))

forall p n r s. (~((p) = 1) /\ forall pfa_factor_left_multiple_unique_domain pfa_factor_right_multiple_unique_domain. (p) = pfa_factor_left_multiple_unique_domain * pfa_factor_right_multiple_unique_domain -> pfa_factor_left_multiple_unique_domain = 1 \/ pfa_factor_right_multiple_unique_domain = 1) -> (exists pff_history_code_multiple_unique_first pff_history_scale_multiple_unique_first. (((((exists ff_h_pft_multiple_unique_firsthistorystart. ff_h_pft_multiple_unique_firsthistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystart. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystart * S ((S (0)) * pff_history_scale_multiple_unique_first) + (0))) /\ (((((exists ff_h_pft_multiple_unique_firsthistoryterminal. ff_h_pft_multiple_unique_firsthistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistoryterminal. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistoryterminal * S ((S (n)) * pff_history_scale_multiple_unique_first) + (r))) /\ ((forall pff_trace_index_multiple_unique_firsthistorysteps. (exists pfa_gap_multiple_unique_firsthistorystepsindex. pfa_gap_multiple_unique_firsthistorystepsindex + S (pff_trace_index_multiple_unique_firsthistorysteps) = (n)) -> exists pff_trace_before_multiple_unique_firsthistorysteps pff_trace_after_multiple_unique_firsthistorysteps. ((((exists ff_h_pft_multiple_unique_firsthistorystepsbefore. ff_h_pft_multiple_unique_firsthistorystepsbefore + S (pff_trace_before_multiple_unique_firsthistorysteps) = S ((S (pff_trace_index_multiple_unique_firsthistorysteps)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystepsbefore. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystepsbefore * S ((S (pff_trace_index_multiple_unique_firsthistorysteps)) * pff_history_scale_multiple_unique_first) + (pff_trace_before_multiple_unique_firsthistorysteps))) /\ (((((exists ff_h_pft_multiple_unique_firsthistorystepsafter. ff_h_pft_multiple_unique_firsthistorystepsafter + S (pff_trace_after_multiple_unique_firsthistorysteps) = S ((S (S (pff_trace_index_multiple_unique_firsthistorysteps))) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystepsafter. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystepsafter * S ((S (S (pff_trace_index_multiple_unique_firsthistorysteps))) * pff_history_scale_multiple_unique_first) + (pff_trace_after_multiple_unique_firsthistorysteps))) /\ ((((exists pfa_gap_multiple_unique_firsthistorystepsadditionleft. pfa_gap_multiple_unique_firsthistorystepsadditionleft + S (pff_trace_before_multiple_unique_firsthistorysteps) = (p)) /\ (((exists pfa_gap_multiple_unique_firsthistorystepsadditionright. pfa_gap_multiple_unique_firsthistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_unique_firsthistorystepsadditionresultbound. pfa_gap_multiple_unique_firsthistorystepsadditionresultbound + S (pff_trace_after_multiple_unique_firsthistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_unique_firsthistorystepsadditionresultcongruence pfa_offset_right_multiple_unique_firsthistorystepsadditionresultcongruence. ((pff_trace_before_multiple_unique_firsthistorysteps) + (1)) + (p) * pfa_offset_left_multiple_unique_firsthistorystepsadditionresultcongruence = (pff_trace_after_multiple_unique_firsthistorysteps) + (p) * pfa_offset_right_multiple_unique_firsthistorystepsadditionresultcongruence)))))))))))))))))))) -> (exists pff_history_code_multiple_unique_second pff_history_scale_multiple_unique_second. (((((exists ff_h_pft_multiple_unique_secondhistorystart. ff_h_pft_multiple_unique_secondhistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystart. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystart * S ((S (0)) * pff_history_scale_multiple_unique_second) + (0))) /\ (((((exists ff_h_pft_multiple_unique_secondhistoryterminal. ff_h_pft_multiple_unique_secondhistoryterminal + S (s) = S ((S (n)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistoryterminal. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistoryterminal * S ((S (n)) * pff_history_scale_multiple_unique_second) + (s))) /\ ((forall pff_trace_index_multiple_unique_secondhistorysteps. (exists pfa_gap_multiple_unique_secondhistorystepsindex. pfa_gap_multiple_unique_secondhistorystepsindex + S (pff_trace_index_multiple_unique_secondhistorysteps) = (n)) -> exists pff_trace_before_multiple_unique_secondhistorysteps pff_trace_after_multiple_unique_secondhistorysteps. ((((exists ff_h_pft_multiple_unique_secondhistorystepsbefore. ff_h_pft_multiple_unique_secondhistorystepsbefore + S (pff_trace_before_multiple_unique_secondhistorysteps) = S ((S (pff_trace_index_multiple_unique_secondhistorysteps)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystepsbefore. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystepsbefore * S ((S (pff_trace_index_multiple_unique_secondhistorysteps)) * pff_history_scale_multiple_unique_second) + (pff_trace_before_multiple_unique_secondhistorysteps))) /\ (((((exists ff_h_pft_multiple_unique_secondhistorystepsafter. ff_h_pft_multiple_unique_secondhistorystepsafter + S (pff_trace_after_multiple_unique_secondhistorysteps) = S ((S (S (pff_trace_index_multiple_unique_secondhistorysteps))) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystepsafter. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystepsafter * S ((S (S (pff_trace_index_multiple_unique_secondhistorysteps))) * pff_history_scale_multiple_unique_second) + (pff_trace_after_multiple_unique_secondhistorysteps))) /\ ((((exists pfa_gap_multiple_unique_secondhistorystepsadditionleft. pfa_gap_multiple_unique_secondhistorystepsadditionleft + S (pff_trace_before_multiple_unique_secondhistorysteps) = (p)) /\ (((exists pfa_gap_multiple_unique_secondhistorystepsadditionright. pfa_gap_multiple_unique_secondhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_unique_secondhistorystepsadditionresultbound. pfa_gap_multiple_unique_secondhistorystepsadditionresultbound + S (pff_trace_after_multiple_unique_secondhistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_unique_secondhistorystepsadditionresultcongruence pfa_offset_right_multiple_unique_secondhistorystepsadditionresultcongruence. ((pff_trace_before_multiple_unique_secondhistorysteps) + (1)) + (p) * pfa_offset_left_multiple_unique_secondhistorystepsadditionresultcongruence = (pff_trace_after_multiple_unique_secondhistorysteps) + (p) * pfa_offset_right_multiple_unique_secondhistorystepsadditionresultcongruence)))))))))))))))))))) -> r = s

forall p n. (~((p) = 1) /\ forall pfa_factor_left_multiple_total_domain pfa_factor_right_multiple_total_domain. (p) = pfa_factor_left_multiple_total_domain * pfa_factor_right_multiple_total_domain -> pfa_factor_left_multiple_total_domain = 1 \/ pfa_factor_right_multiple_total_domain = 1) -> exists r. (exists pff_history_code_multiple_total_chosen pff_history_scale_multiple_total_chosen. (((((exists ff_h_pft_multiple_total_chosenhistorystart. ff_h_pft_multiple_total_chosenhistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_total_chosen)) /\ exists ff_q_pft_multiple_total_chosenhistorystart. pff_history_code_multiple_total_chosen = ff_q_pft_multiple_total_chosenhistorystart * S ((S (0)) * pff_history_scale_multiple_total_chosen) + (0))) /\ (((((exists ff_h_pft_multiple_total_chosenhistoryterminal. ff_h_pft_multiple_total_chosenhistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_total_chosen)) /\ exists ff_q_pft_multiple_total_chosenhistoryterminal. pff_history_code_multiple_total_chosen = ff_q_pft_multiple_total_chosenhistoryterminal * S ((S (n)) * pff_history_scale_multiple_total_chosen) + (r))) /\ ((forall pff_trace_index_multiple_total_chosenhistorysteps. (exists pfa_gap_multiple_total_chosenhistorystepsindex. pfa_gap_multiple_total_chosenhistorystepsindex + S (pff_trace_index_multiple_total_chosenhistorysteps) = (n)) -> exists pff_trace_before_multiple_total_chosenhistorysteps pff_trace_after_multiple_total_chosenhistorysteps. ((((exists ff_h_pft_multiple_total_chosenhistorystepsbefore. ff_h_pft_multiple_total_chosenhistorystepsbefore + S (pff_trace_before_multiple_total_chosenhistorysteps) = S ((S (pff_trace_index_multiple_total_chosenhistorysteps)) * pff_history_scale_multiple_total_chosen)) /\ exists ff_q_pft_multiple_total_chosenhistorystepsbefore. pff_history_code_multiple_total_chosen = ff_q_pft_multiple_total_chosenhistorystepsbefore * S ((S (pff_trace_index_multiple_total_chosenhistorysteps)) * pff_history_scale_multiple_total_chosen) + (pff_trace_before_multiple_total_chosenhistorysteps))) /\ (((((exists ff_h_pft_multiple_total_chosenhistorystepsafter. ff_h_pft_multiple_total_chosenhistorystepsafter + S (pff_trace_after_multiple_total_chosenhistorysteps) = S ((S (S (pff_trace_index_multiple_total_chosenhistorysteps))) * pff_history_scale_multiple_total_chosen)) /\ exists ff_q_pft_multiple_total_chosenhistorystepsafter. pff_history_code_multiple_total_chosen = ff_q_pft_multiple_total_chosenhistorystepsafter * S ((S (S (pff_trace_index_multiple_total_chosenhistorysteps))) * pff_history_scale_multiple_total_chosen) + (pff_trace_after_multiple_total_chosenhistorysteps))) /\ ((((exists pfa_gap_multiple_total_chosenhistorystepsadditionleft. pfa_gap_multiple_total_chosenhistorystepsadditionleft + S (pff_trace_before_multiple_total_chosenhistorysteps) = (p)) /\ (((exists pfa_gap_multiple_total_chosenhistorystepsadditionright. pfa_gap_multiple_total_chosenhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_total_chosenhistorystepsadditionresultbound. pfa_gap_multiple_total_chosenhistorystepsadditionresultbound + S (pff_trace_after_multiple_total_chosenhistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_total_chosenhistorystepsadditionresultcongruence pfa_offset_right_multiple_total_chosenhistorystepsadditionresultcongruence. ((pff_trace_before_multiple_total_chosenhistorysteps) + (1)) + (p) * pfa_offset_left_multiple_total_chosenhistorystepsadditionresultcongruence = (pff_trace_after_multiple_total_chosenhistorysteps) + (p) * pfa_offset_right_multiple_total_chosenhistorystepsadditionresultcongruence)))))))))))))))))))) /\ forall s. (exists pff_history_code_multiple_total_other pff_history_scale_multiple_total_other. (((((exists ff_h_pft_multiple_total_otherhistorystart. ff_h_pft_multiple_total_otherhistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_total_other)) /\ exists ff_q_pft_multiple_total_otherhistorystart. pff_history_code_multiple_total_other = ff_q_pft_multiple_total_otherhistorystart * S ((S (0)) * pff_history_scale_multiple_total_other) + (0))) /\ (((((exists ff_h_pft_multiple_total_otherhistoryterminal. ff_h_pft_multiple_total_otherhistoryterminal + S (s) = S ((S (n)) * pff_history_scale_multiple_total_other)) /\ exists ff_q_pft_multiple_total_otherhistoryterminal. pff_history_code_multiple_total_other = ff_q_pft_multiple_total_otherhistoryterminal * S ((S (n)) * pff_history_scale_multiple_total_other) + (s))) /\ ((forall pff_trace_index_multiple_total_otherhistorysteps. (exists pfa_gap_multiple_total_otherhistorystepsindex. pfa_gap_multiple_total_otherhistorystepsindex + S (pff_trace_index_multiple_total_otherhistorysteps) = (n)) -> exists pff_trace_before_multiple_total_otherhistorysteps pff_trace_after_multiple_total_otherhistorysteps. ((((exists ff_h_pft_multiple_total_otherhistorystepsbefore. ff_h_pft_multiple_total_otherhistorystepsbefore + S (pff_trace_before_multiple_total_otherhistorysteps) = S ((S (pff_trace_index_multiple_total_otherhistorysteps)) * pff_history_scale_multiple_total_other)) /\ exists ff_q_pft_multiple_total_otherhistorystepsbefore. pff_history_code_multiple_total_other = ff_q_pft_multiple_total_otherhistorystepsbefore * S ((S (pff_trace_index_multiple_total_otherhistorysteps)) * pff_history_scale_multiple_total_other) + (pff_trace_before_multiple_total_otherhistorysteps))) /\ (((((exists ff_h_pft_multiple_total_otherhistorystepsafter. ff_h_pft_multiple_total_otherhistorystepsafter + S (pff_trace_after_multiple_total_otherhistorysteps) = S ((S (S (pff_trace_index_multiple_total_otherhistorysteps))) * pff_history_scale_multiple_total_other)) /\ exists ff_q_pft_multiple_total_otherhistorystepsafter. pff_history_code_multiple_total_other = ff_q_pft_multiple_total_otherhistorystepsafter * S ((S (S (pff_trace_index_multiple_total_otherhistorysteps))) * pff_history_scale_multiple_total_other) + (pff_trace_after_multiple_total_otherhistorysteps))) /\ ((((exists pfa_gap_multiple_total_otherhistorystepsadditionleft. pfa_gap_multiple_total_otherhistorystepsadditionleft + S (pff_trace_before_multiple_total_otherhistorysteps) = (p)) /\ (((exists pfa_gap_multiple_total_otherhistorystepsadditionright. pfa_gap_multiple_total_otherhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_total_otherhistorystepsadditionresultbound. pfa_gap_multiple_total_otherhistorystepsadditionresultbound + S (pff_trace_after_multiple_total_otherhistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_total_otherhistorystepsadditionresultcongruence pfa_offset_right_multiple_total_otherhistorystepsadditionresultcongruence. ((pff_trace_before_multiple_total_otherhistorysteps) + (1)) + (p) * pfa_offset_left_multiple_total_otherhistorystepsadditionresultcongruence = (pff_trace_after_multiple_total_otherhistorysteps) + (p) * pfa_offset_right_multiple_total_otherhistorystepsadditionresultcongruence)))))))))))))))))))) -> s = r

forall p. (~((p) = 1) /\ forall pfa_factor_left_characteristic_domain pfa_factor_right_characteristic_domain. (p) = pfa_factor_left_characteristic_domain * pfa_factor_right_characteristic_domain -> pfa_factor_left_characteristic_domain = 1 \/ pfa_factor_right_characteristic_domain = 1) -> (((exists pff_history_code_characteristic_exactmodulus pff_history_scale_characteristic_exactmodulus. (((((exists ff_h_pft_characteristic_exactmodulushistorystart. ff_h_pft_characteristic_exactmodulushistorystart + S (0) = S ((S (0)) * pff_history_scale_characteristic_exactmodulus)) /\ exists ff_q_pft_characteristic_exactmodulushistorystart. pff_history_code_characteristic_exactmodulus = ff_q_pft_characteristic_exactmodulushistorystart * S ((S (0)) * pff_history_scale_characteristic_exactmodulus) + (0))) /\ (((((exists ff_h_pft_characteristic_exactmodulushistoryterminal. ff_h_pft_characteristic_exactmodulushistoryterminal + S (0) = S ((S (p)) * pff_history_scale_characteristic_exactmodulus)) /\ exists ff_q_pft_characteristic_exactmodulushistoryterminal. pff_history_code_characteristic_exactmodulus = ff_q_pft_characteristic_exactmodulushistoryterminal * S ((S (p)) * pff_history_scale_characteristic_exactmodulus) + (0))) /\ ((forall pff_trace_index_characteristic_exactmodulushistorysteps. (exists pfa_gap_characteristic_exactmodulushistorystepsindex. pfa_gap_characteristic_exactmodulushistorystepsindex + S (pff_trace_index_characteristic_exactmodulushistorysteps) = (p)) -> exists pff_trace_before_characteristic_exactmodulushistorysteps pff_trace_after_characteristic_exactmodulushistorysteps. ((((exists ff_h_pft_characteristic_exactmodulushistorystepsbefore. ff_h_pft_characteristic_exactmodulushistorystepsbefore + S (pff_trace_before_characteristic_exactmodulushistorysteps) = S ((S (pff_trace_index_characteristic_exactmodulushistorysteps)) * pff_history_scale_characteristic_exactmodulus)) /\ exists ff_q_pft_characteristic_exactmodulushistorystepsbefore. pff_history_code_characteristic_exactmodulus = ff_q_pft_characteristic_exactmodulushistorystepsbefore * S ((S (pff_trace_index_characteristic_exactmodulushistorysteps)) * pff_history_scale_characteristic_exactmodulus) + (pff_trace_before_characteristic_exactmodulushistorysteps))) /\ (((((exists ff_h_pft_characteristic_exactmodulushistorystepsafter. ff_h_pft_characteristic_exactmodulushistorystepsafter + S (pff_trace_after_characteristic_exactmodulushistorysteps) = S ((S (S (pff_trace_index_characteristic_exactmodulushistorysteps))) * pff_history_scale_characteristic_exactmodulus)) /\ exists ff_q_pft_characteristic_exactmodulushistorystepsafter. pff_history_code_characteristic_exactmodulus = ff_q_pft_characteristic_exactmodulushistorystepsafter * S ((S (S (pff_trace_index_characteristic_exactmodulushistorysteps))) * pff_history_scale_characteristic_exactmodulus) + (pff_trace_after_characteristic_exactmodulushistorysteps))) /\ ((((exists pfa_gap_characteristic_exactmodulushistorystepsadditionleft. pfa_gap_characteristic_exactmodulushistorystepsadditionleft + S (pff_trace_before_characteristic_exactmodulushistorysteps) = (p)) /\ (((exists pfa_gap_characteristic_exactmodulushistorystepsadditionright. pfa_gap_characteristic_exactmodulushistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_characteristic_exactmodulushistorystepsadditionresultbound. pfa_gap_characteristic_exactmodulushistorystepsadditionresultbound + S (pff_trace_after_characteristic_exactmodulushistorysteps) = (p)) /\ ((exists pfa_offset_left_characteristic_exactmodulushistorystepsadditionresultcongruence pfa_offset_right_characteristic_exactmodulushistorystepsadditionresultcongruence. ((pff_trace_before_characteristic_exactmodulushistorysteps) + (1)) + (p) * pfa_offset_left_characteristic_exactmodulushistorystepsadditionresultcongruence = (pff_trace_after_characteristic_exactmodulushistorysteps) + (p) * pfa_offset_right_characteristic_exactmodulushistorystepsadditionresultcongruence)))))))))))))))))))) /\ ((forall pff_smaller_positive_characteristic_exact. (exists pfa_gap_characteristic_exactstrict. pfa_gap_characteristic_exactstrict + S (pff_smaller_positive_characteristic_exact) = (p)) -> ~(pff_smaller_positive_characteristic_exact = 0) -> ~(exists pff_history_code_characteristic_exactsmaller pff_history_scale_characteristic_exactsmaller. (((((exists ff_h_pft_characteristic_exactsmallerhistorystart. ff_h_pft_characteristic_exactsmallerhistorystart + S (0) = S ((S (0)) * pff_history_scale_characteristic_exactsmaller)) /\ exists ff_q_pft_characteristic_exactsmallerhistorystart. pff_history_code_characteristic_exactsmaller = ff_q_pft_characteristic_exactsmallerhistorystart * S ((S (0)) * pff_history_scale_characteristic_exactsmaller) + (0))) /\ (((((exists ff_h_pft_characteristic_exactsmallerhistoryterminal. ff_h_pft_characteristic_exactsmallerhistoryterminal + S (0) = S ((S (pff_smaller_positive_characteristic_exact)) * pff_history_scale_characteristic_exactsmaller)) /\ exists ff_q_pft_characteristic_exactsmallerhistoryterminal. pff_history_code_characteristic_exactsmaller = ff_q_pft_characteristic_exactsmallerhistoryterminal * S ((S (pff_smaller_positive_characteristic_exact)) * pff_history_scale_characteristic_exactsmaller) + (0))) /\ ((forall pff_trace_index_characteristic_exactsmallerhistorysteps. (exists pfa_gap_characteristic_exactsmallerhistorystepsindex. pfa_gap_characteristic_exactsmallerhistorystepsindex + S (pff_trace_index_characteristic_exactsmallerhistorysteps) = (pff_smaller_positive_characteristic_exact)) -> exists pff_trace_before_characteristic_exactsmallerhistorysteps pff_trace_after_characteristic_exactsmallerhistorysteps. ((((exists ff_h_pft_characteristic_exactsmallerhistorystepsbefore. ff_h_pft_characteristic_exactsmallerhistorystepsbefore + S (pff_trace_before_characteristic_exactsmallerhistorysteps) = S ((S (pff_trace_index_characteristic_exactsmallerhistorysteps)) * pff_history_scale_characteristic_exactsmaller)) /\ exists ff_q_pft_characteristic_exactsmallerhistorystepsbefore. pff_history_code_characteristic_exactsmaller = ff_q_pft_characteristic_exactsmallerhistorystepsbefore * S ((S (pff_trace_index_characteristic_exactsmallerhistorysteps)) * pff_history_scale_characteristic_exactsmaller) + (pff_trace_before_characteristic_exactsmallerhistorysteps))) /\ (((((exists ff_h_pft_characteristic_exactsmallerhistorystepsafter. ff_h_pft_characteristic_exactsmallerhistorystepsafter + S (pff_trace_after_characteristic_exactsmallerhistorysteps) = S ((S (S (pff_trace_index_characteristic_exactsmallerhistorysteps))) * pff_history_scale_characteristic_exactsmaller)) /\ exists ff_q_pft_characteristic_exactsmallerhistorystepsafter. pff_history_code_characteristic_exactsmaller = ff_q_pft_characteristic_exactsmallerhistorystepsafter * S ((S (S (pff_trace_index_characteristic_exactsmallerhistorysteps))) * pff_history_scale_characteristic_exactsmaller) + (pff_trace_after_characteristic_exactsmallerhistorysteps))) /\ ((((exists pfa_gap_characteristic_exactsmallerhistorystepsadditionleft. pfa_gap_characteristic_exactsmallerhistorystepsadditionleft + S (pff_trace_before_characteristic_exactsmallerhistorysteps) = (p)) /\ (((exists pfa_gap_characteristic_exactsmallerhistorystepsadditionright. pfa_gap_characteristic_exactsmallerhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_characteristic_exactsmallerhistorystepsadditionresultbound. pfa_gap_characteristic_exactsmallerhistorystepsadditionresultbound + S (pff_trace_after_characteristic_exactsmallerhistorysteps) = (p)) /\ ((exists pfa_offset_left_characteristic_exactsmallerhistorystepsadditionresultcongruence pfa_offset_right_characteristic_exactsmallerhistorystepsadditionresultcongruence. ((pff_trace_before_characteristic_exactsmallerhistorysteps) + (1)) + (p) * pfa_offset_left_characteristic_exactsmallerhistorystepsadditionresultcongruence = (pff_trace_after_characteristic_exactsmallerhistorysteps) + (p) * pfa_offset_right_characteristic_exactsmallerhistorystepsadditionresultcongruence))))))))))))))))))))))))

forall p. (~((p) = 1) /\ forall pfa_factor_left_finite_structure_domain pfa_factor_right_finite_structure_domain. (p) = pfa_factor_left_finite_structure_domain * pfa_factor_right_finite_structure_domain -> pfa_factor_left_finite_structure_domain = 1 \/ pfa_factor_right_finite_structure_domain = 1) -> exists ab ac mb mc nb nc ib ic eb ec. (((((forall pft_index_finite_structuretablesadd. (exists pfa_gap_finite_structuretablesaddprefix. pfa_gap_finite_structuretablesaddprefix + S (pft_index_finite_structuretablesadd) = ((p) * (p))) -> exists pft_value_finite_structuretablesadd. (((((exists ff_h_pft_finite_structuretablesaddpointentry. ff_h_pft_finite_structuretablesaddpointentry + S (pft_value_finite_structuretablesadd) = S ((S (pft_index_finite_structuretablesadd)) * ac)) /\ exists ff_q_pft_finite_structuretablesaddpointentry. ab = ff_q_pft_finite_structuretablesaddpointentry * S ((S (pft_index_finite_structuretablesadd)) * ac) + (pft_value_finite_structuretablesadd))) /\ ((exists pft_row_finite_structuretablesaddpointvalue pft_column_finite_structuretablesaddpointvalue. (((pft_index_finite_structuretablesadd) = pft_row_finite_structuretablesaddpointvalue * (p) + pft_column_finite_structuretablesaddpointvalue) /\ ((((exists pfa_gap_finite_structuretablesaddpointvalueoperationleft. pfa_gap_finite_structuretablesaddpointvalueoperationleft + S (pft_row_finite_structuretablesaddpointvalue) = (p)) /\ (((exists pfa_gap_finite_structuretablesaddpointvalueoperationright. pfa_gap_finite_structuretablesaddpointvalueoperationright + S (pft_column_finite_structuretablesaddpointvalue) = (p)) /\ ((((exists pfa_gap_finite_structuretablesaddpointvalueoperationresultbound. pfa_gap_finite_structuretablesaddpointvalueoperationresultbound + S (pft_value_finite_structuretablesadd) = (p)) /\ ((exists pfa_offset_left_finite_structuretablesaddpointvalueoperationresultcongruence pfa_offset_right_finite_structuretablesaddpointvalueoperationresultcongruence. ((pft_row_finite_structuretablesaddpointvalue) + (pft_column_finite_structuretablesaddpointvalue)) + (p) * pfa_offset_left_finite_structuretablesaddpointvalueoperationresultcongruence = (pft_value_finite_structuretablesadd) + (p) * pfa_offset_right_finite_structuretablesaddpointvalueoperationresultcongruence)))))))))))))))) /\ (((forall pft_index_finite_structuretablesmultiply. (exists pfa_gap_finite_structuretablesmultiplyprefix. pfa_gap_finite_structuretablesmultiplyprefix + S (pft_index_finite_structuretablesmultiply) = ((p) * (p))) -> exists pft_value_finite_structuretablesmultiply. (((((exists ff_h_pft_finite_structuretablesmultiplypointentry. ff_h_pft_finite_structuretablesmultiplypointentry + S (pft_value_finite_structuretablesmultiply) = S ((S (pft_index_finite_structuretablesmultiply)) * mc)) /\ exists ff_q_pft_finite_structuretablesmultiplypointentry. mb = ff_q_pft_finite_structuretablesmultiplypointentry * S ((S (pft_index_finite_structuretablesmultiply)) * mc) + (pft_value_finite_structuretablesmultiply))) /\ ((exists pft_row_finite_structuretablesmultiplypointvalue pft_column_finite_structuretablesmultiplypointvalue. (((pft_index_finite_structuretablesmultiply) = pft_row_finite_structuretablesmultiplypointvalue * (p) + pft_column_finite_structuretablesmultiplypointvalue) /\ ((((exists pfa_gap_finite_structuretablesmultiplypointvalueoperationleft. pfa_gap_finite_structuretablesmultiplypointvalueoperationleft + S (pft_row_finite_structuretablesmultiplypointvalue) = (p)) /\ (((exists pfa_gap_finite_structuretablesmultiplypointvalueoperationright. pfa_gap_finite_structuretablesmultiplypointvalueoperationright + S (pft_column_finite_structuretablesmultiplypointvalue) = (p)) /\ ((((exists pfa_gap_finite_structuretablesmultiplypointvalueoperationresultbound. pfa_gap_finite_structuretablesmultiplypointvalueoperationresultbound + S (pft_value_finite_structuretablesmultiply) = (p)) /\ ((exists pfa_offset_left_finite_structuretablesmultiplypointvalueoperationresultcongruence pfa_offset_right_finite_structuretablesmultiplypointvalueoperationresultcongruence. ((pft_row_finite_structuretablesmultiplypointvalue) * (pft_column_finite_structuretablesmultiplypointvalue)) + (p) * pfa_offset_left_finite_structuretablesmultiplypointvalueoperationresultcongruence = (pft_value_finite_structuretablesmultiply) + (p) * pfa_offset_right_finite_structuretablesmultiplypointvalueoperationresultcongruence)))))))))))))))) /\ (((forall pft_index_finite_structuretablesnegate. (exists pfa_gap_finite_structuretablesnegateprefix. pfa_gap_finite_structuretablesnegateprefix + S (pft_index_finite_structuretablesnegate) = (p)) -> exists pft_value_finite_structuretablesnegate. (((((exists ff_h_pft_finite_structuretablesnegatepointentry. ff_h_pft_finite_structuretablesnegatepointentry + S (pft_value_finite_structuretablesnegate) = S ((S (pft_index_finite_structuretablesnegate)) * nc)) /\ exists ff_q_pft_finite_structuretablesnegatepointentry. nb = ff_q_pft_finite_structuretablesnegatepointentry * S ((S (pft_index_finite_structuretablesnegate)) * nc) + (pft_value_finite_structuretablesnegate))) /\ ((((exists pfa_gap_finite_structuretablesnegatepointvalueadditionleft. pfa_gap_finite_structuretablesnegatepointvalueadditionleft + S (pft_index_finite_structuretablesnegate) = (p)) /\ (((exists pfa_gap_finite_structuretablesnegatepointvalueadditionright. pfa_gap_finite_structuretablesnegatepointvalueadditionright + S (pft_value_finite_structuretablesnegate) = (p)) /\ ((((exists pfa_gap_finite_structuretablesnegatepointvalueadditionresultbound. pfa_gap_finite_structuretablesnegatepointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structuretablesnegatepointvalueadditionresultcongruence pfa_offset_right_finite_structuretablesnegatepointvalueadditionresultcongruence. ((pft_index_finite_structuretablesnegate) + (pft_value_finite_structuretablesnegate)) + (p) * pfa_offset_left_finite_structuretablesnegatepointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_finite_structuretablesnegatepointvalueadditionresultcongruence))))))))))))) /\ ((forall pft_index_finite_structuretablesinverse. (exists pfa_gap_finite_structuretablesinverseprefix. pfa_gap_finite_structuretablesinverseprefix + S (pft_index_finite_structuretablesinverse) = (p)) -> exists pft_value_finite_structuretablesinverse. (((((exists ff_h_pft_finite_structuretablesinversepointentry. ff_h_pft_finite_structuretablesinversepointentry + S (pft_value_finite_structuretablesinverse) = S ((S (pft_index_finite_structuretablesinverse)) * ic)) /\ exists ff_q_pft_finite_structuretablesinversepointentry. ib = ff_q_pft_finite_structuretablesinversepointentry * S ((S (pft_index_finite_structuretablesinverse)) * ic) + (pft_value_finite_structuretablesinverse))) /\ ((((exists pfa_gap_finite_structuretablesinversepointvalueinput. pfa_gap_finite_structuretablesinversepointvalueinput + S (pft_index_finite_structuretablesinverse) = (p)) /\ (((exists pfa_gap_finite_structuretablesinversepointvalueoutput. pfa_gap_finite_structuretablesinversepointvalueoutput + S (pft_value_finite_structuretablesinverse) = (p)) /\ ((((pft_index_finite_structuretablesinverse) = 0 /\ (pft_value_finite_structuretablesinverse) = 0) \/ (((~((pft_index_finite_structuretablesinverse) = 0)) /\ ((((exists pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationleft. pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationleft + S (pft_index_finite_structuretablesinverse) = (p)) /\ (((exists pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationright. pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationright + S (pft_value_finite_structuretablesinverse) = (p)) /\ ((((exists pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationresultbound. pfa_gap_finite_structuretablesinversepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_finite_structuretablesinversepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_finite_structuretablesinversepointvaluenonzeromultiplicationresultcongruence. ((pft_index_finite_structuretablesinverse) * (pft_value_finite_structuretablesinverse)) + (p) * pfa_offset_left_finite_structuretablesinversepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_finite_structuretablesinversepointvaluenonzeromultiplicationresultcongruence))))))))))))))))))))))))))))) /\ (((((forall pff_enumeration_index_finite_structurecardinalityenumeration. (exists pfa_gap_finite_structurecardinalityenumerationbound. pfa_gap_finite_structurecardinalityenumerationbound + S (pff_enumeration_index_finite_structurecardinalityenumeration) = (p)) -> (((exists ff_h_pft_finite_structurecardinalityenumerationentry. ff_h_pft_finite_structurecardinalityenumerationentry + S (pff_enumeration_index_finite_structurecardinalityenumeration) = S ((S (pff_enumeration_index_finite_structurecardinalityenumeration)) * ec)) /\ exists ff_q_pft_finite_structurecardinalityenumerationentry. eb = ff_q_pft_finite_structurecardinalityenumerationentry * S ((S (pff_enumeration_index_finite_structurecardinalityenumeration)) * ec) + (pff_enumeration_index_finite_structurecardinalityenumeration)))) /\ (((forall pff_cardinality_i_finite_structurecardinality pff_cardinality_a_finite_structurecardinality. (exists pfa_gap_finite_structurecardinalitybounded_index. pfa_gap_finite_structurecardinalitybounded_index + S (pff_cardinality_i_finite_structurecardinality) = (p)) -> (((exists ff_h_pft_finite_structurecardinalitybounded_entry. ff_h_pft_finite_structurecardinalitybounded_entry + S (pff_cardinality_a_finite_structurecardinality) = S ((S (pff_cardinality_i_finite_structurecardinality)) * ec)) /\ exists ff_q_pft_finite_structurecardinalitybounded_entry. eb = ff_q_pft_finite_structurecardinalitybounded_entry * S ((S (pff_cardinality_i_finite_structurecardinality)) * ec) + (pff_cardinality_a_finite_structurecardinality))) -> (exists pfa_gap_finite_structurecardinalitybounded_value. pfa_gap_finite_structurecardinalitybounded_value + S (pff_cardinality_a_finite_structurecardinality) = (p))) /\ (((forall pff_cardinality_i_finite_structurecardinality pff_cardinality_j_finite_structurecardinality pff_cardinality_a_finite_structurecardinality. (exists pfa_gap_finite_structurecardinalityinjective_i. pfa_gap_finite_structurecardinalityinjective_i + S (pff_cardinality_i_finite_structurecardinality) = (p)) -> (exists pfa_gap_finite_structurecardinalityinjective_j. pfa_gap_finite_structurecardinalityinjective_j + S (pff_cardinality_j_finite_structurecardinality) = (p)) -> (((exists ff_h_pft_finite_structurecardinalityinjective_first. ff_h_pft_finite_structurecardinalityinjective_first + S (pff_cardinality_a_finite_structurecardinality) = S ((S (pff_cardinality_i_finite_structurecardinality)) * ec)) /\ exists ff_q_pft_finite_structurecardinalityinjective_first. eb = ff_q_pft_finite_structurecardinalityinjective_first * S ((S (pff_cardinality_i_finite_structurecardinality)) * ec) + (pff_cardinality_a_finite_structurecardinality))) -> (((exists ff_h_pft_finite_structurecardinalityinjective_second. ff_h_pft_finite_structurecardinalityinjective_second + S (pff_cardinality_a_finite_structurecardinality) = S ((S (pff_cardinality_j_finite_structurecardinality)) * ec)) /\ exists ff_q_pft_finite_structurecardinalityinjective_second. eb = ff_q_pft_finite_structurecardinalityinjective_second * S ((S (pff_cardinality_j_finite_structurecardinality)) * ec) + (pff_cardinality_a_finite_structurecardinality))) -> pff_cardinality_i_finite_structurecardinality = pff_cardinality_j_finite_structurecardinality) /\ ((forall pff_cardinality_a_finite_structurecardinality. (exists pfa_gap_finite_structurecardinalitysurjective_value. pfa_gap_finite_structurecardinalitysurjective_value + S (pff_cardinality_a_finite_structurecardinality) = (p)) -> exists pff_cardinality_i_finite_structurecardinality. (exists pfa_gap_finite_structurecardinalitysurjective_index. pfa_gap_finite_structurecardinalitysurjective_index + S (pff_cardinality_i_finite_structurecardinality) = (p)) /\ (((exists ff_h_pft_finite_structurecardinalitysurjective_entry. ff_h_pft_finite_structurecardinalitysurjective_entry + S (pff_cardinality_a_finite_structurecardinality) = S ((S (pff_cardinality_i_finite_structurecardinality)) * ec)) /\ exists ff_q_pft_finite_structurecardinalitysurjective_entry. eb = ff_q_pft_finite_structurecardinalitysurjective_entry * S ((S (pff_cardinality_i_finite_structurecardinality)) * ec) + (pff_cardinality_a_finite_structurecardinality))))))))))) /\ (((((exists pfa_gap_finite_structurelawszero. pfa_gap_finite_structurelawszero + S (0) = (p)) /\ (((exists pfa_gap_finite_structurelawsone. pfa_gap_finite_structurelawsone + S (1) = (p)) /\ (((~(0 = 1)) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws. (exists pfa_gap_finite_structurelawsaddleft. pfa_gap_finite_structurelawsaddleft + S (pfa_law_a_finite_structurelaws) = (p)) -> (exists pfa_gap_finite_structurelawsaddright. pfa_gap_finite_structurelawsaddright + S (pfa_law_b_finite_structurelaws) = (p)) -> exists pfa_law_c_finite_structurelaws. (((exists pfa_gap_finite_structurelawsaddchosenleft. pfa_gap_finite_structurelawsaddchosenleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddchosenright. pfa_gap_finite_structurelawsaddchosenright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddchosenresultbound. pfa_gap_finite_structurelawsaddchosenresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddchosenresultcongruence pfa_offset_right_finite_structurelawsaddchosenresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddchosenresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddchosenresultcongruence))))))))) /\ forall pfa_law_d_finite_structurelaws. (((exists pfa_gap_finite_structurelawsaddotherleft. pfa_gap_finite_structurelawsaddotherleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddotherright. pfa_gap_finite_structurelawsaddotherright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddotherresultbound. pfa_gap_finite_structurelawsaddotherresultbound + S (pfa_law_d_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddotherresultcongruence pfa_offset_right_finite_structurelawsaddotherresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddotherresultcongruence = (pfa_law_d_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddotherresultcongruence))))))))) -> pfa_law_d_finite_structurelaws = pfa_law_c_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws. (((exists pfa_gap_finite_structurelawsaddcomm_firstleft. pfa_gap_finite_structurelawsaddcomm_firstleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddcomm_firstright. pfa_gap_finite_structurelawsaddcomm_firstright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddcomm_firstresultbound. pfa_gap_finite_structurelawsaddcomm_firstresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddcomm_firstresultcongruence pfa_offset_right_finite_structurelawsaddcomm_firstresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddcomm_firstresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddcomm_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsaddcomm_secondleft. pfa_gap_finite_structurelawsaddcomm_secondleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddcomm_secondright. pfa_gap_finite_structurelawsaddcomm_secondright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddcomm_secondresultbound. pfa_gap_finite_structurelawsaddcomm_secondresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddcomm_secondresultcongruence pfa_offset_right_finite_structurelawsaddcomm_secondresultcongruence. ((pfa_law_b_finite_structurelaws) + (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddcomm_secondresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddcomm_secondresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws pfa_law_x_finite_structurelaws pfa_law_y_finite_structurelaws pfa_law_u_finite_structurelaws pfa_law_v_finite_structurelaws. (((exists pfa_gap_finite_structurelawsaddassoc_firstleft. pfa_gap_finite_structurelawsaddassoc_firstleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddassoc_firstright. pfa_gap_finite_structurelawsaddassoc_firstright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddassoc_firstresultbound. pfa_gap_finite_structurelawsaddassoc_firstresultbound + S (pfa_law_x_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddassoc_firstresultcongruence pfa_offset_right_finite_structurelawsaddassoc_firstresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddassoc_firstresultcongruence = (pfa_law_x_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsaddassoc_leftleft. pfa_gap_finite_structurelawsaddassoc_leftleft + S (pfa_law_x_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddassoc_leftright. pfa_gap_finite_structurelawsaddassoc_leftright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddassoc_leftresultbound. pfa_gap_finite_structurelawsaddassoc_leftresultbound + S (pfa_law_u_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddassoc_leftresultcongruence pfa_offset_right_finite_structurelawsaddassoc_leftresultcongruence. ((pfa_law_x_finite_structurelaws) + (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddassoc_leftresultcongruence = (pfa_law_u_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsaddassoc_secondleft. pfa_gap_finite_structurelawsaddassoc_secondleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddassoc_secondright. pfa_gap_finite_structurelawsaddassoc_secondright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddassoc_secondresultbound. pfa_gap_finite_structurelawsaddassoc_secondresultbound + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddassoc_secondresultcongruence pfa_offset_right_finite_structurelawsaddassoc_secondresultcongruence. ((pfa_law_b_finite_structurelaws) + (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddassoc_secondresultcongruence = (pfa_law_y_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsaddassoc_rightleft. pfa_gap_finite_structurelawsaddassoc_rightleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsaddassoc_rightright. pfa_gap_finite_structurelawsaddassoc_rightright + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsaddassoc_rightresultbound. pfa_gap_finite_structurelawsaddassoc_rightresultbound + S (pfa_law_v_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsaddassoc_rightresultcongruence pfa_offset_right_finite_structurelawsaddassoc_rightresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_y_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsaddassoc_rightresultcongruence = (pfa_law_v_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsaddassoc_rightresultcongruence))))))))) -> pfa_law_u_finite_structurelaws = pfa_law_v_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws. (exists pfa_gap_finite_structurelawsmultiplyleft. pfa_gap_finite_structurelawsmultiplyleft + S (pfa_law_a_finite_structurelaws) = (p)) -> (exists pfa_gap_finite_structurelawsmultiplyright. pfa_gap_finite_structurelawsmultiplyright + S (pfa_law_b_finite_structurelaws) = (p)) -> exists pfa_law_c_finite_structurelaws. (((exists pfa_gap_finite_structurelawsmultiplychosenleft. pfa_gap_finite_structurelawsmultiplychosenleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplychosenright. pfa_gap_finite_structurelawsmultiplychosenright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplychosenresultbound. pfa_gap_finite_structurelawsmultiplychosenresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplychosenresultcongruence pfa_offset_right_finite_structurelawsmultiplychosenresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplychosenresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplychosenresultcongruence))))))))) /\ forall pfa_law_d_finite_structurelaws. (((exists pfa_gap_finite_structurelawsmultiplyotherleft. pfa_gap_finite_structurelawsmultiplyotherleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplyotherright. pfa_gap_finite_structurelawsmultiplyotherright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplyotherresultbound. pfa_gap_finite_structurelawsmultiplyotherresultbound + S (pfa_law_d_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplyotherresultcongruence pfa_offset_right_finite_structurelawsmultiplyotherresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplyotherresultcongruence = (pfa_law_d_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplyotherresultcongruence))))))))) -> pfa_law_d_finite_structurelaws = pfa_law_c_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws. (((exists pfa_gap_finite_structurelawsmultiplycomm_firstleft. pfa_gap_finite_structurelawsmultiplycomm_firstleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplycomm_firstright. pfa_gap_finite_structurelawsmultiplycomm_firstright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplycomm_firstresultbound. pfa_gap_finite_structurelawsmultiplycomm_firstresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplycomm_firstresultcongruence pfa_offset_right_finite_structurelawsmultiplycomm_firstresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplycomm_firstresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplycomm_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsmultiplycomm_secondleft. pfa_gap_finite_structurelawsmultiplycomm_secondleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplycomm_secondright. pfa_gap_finite_structurelawsmultiplycomm_secondright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplycomm_secondresultbound. pfa_gap_finite_structurelawsmultiplycomm_secondresultbound + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplycomm_secondresultcongruence pfa_offset_right_finite_structurelawsmultiplycomm_secondresultcongruence. ((pfa_law_b_finite_structurelaws) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplycomm_secondresultcongruence = (pfa_law_c_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplycomm_secondresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws pfa_law_x_finite_structurelaws pfa_law_y_finite_structurelaws pfa_law_u_finite_structurelaws pfa_law_v_finite_structurelaws. (((exists pfa_gap_finite_structurelawsmultiplyassoc_firstleft. pfa_gap_finite_structurelawsmultiplyassoc_firstleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplyassoc_firstright. pfa_gap_finite_structurelawsmultiplyassoc_firstright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplyassoc_firstresultbound. pfa_gap_finite_structurelawsmultiplyassoc_firstresultbound + S (pfa_law_x_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplyassoc_firstresultcongruence pfa_offset_right_finite_structurelawsmultiplyassoc_firstresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplyassoc_firstresultcongruence = (pfa_law_x_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplyassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsmultiplyassoc_leftleft. pfa_gap_finite_structurelawsmultiplyassoc_leftleft + S (pfa_law_x_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplyassoc_leftright. pfa_gap_finite_structurelawsmultiplyassoc_leftright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplyassoc_leftresultbound. pfa_gap_finite_structurelawsmultiplyassoc_leftresultbound + S (pfa_law_u_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplyassoc_leftresultcongruence pfa_offset_right_finite_structurelawsmultiplyassoc_leftresultcongruence. ((pfa_law_x_finite_structurelaws) * (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplyassoc_leftresultcongruence = (pfa_law_u_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplyassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsmultiplyassoc_secondleft. pfa_gap_finite_structurelawsmultiplyassoc_secondleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplyassoc_secondright. pfa_gap_finite_structurelawsmultiplyassoc_secondright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplyassoc_secondresultbound. pfa_gap_finite_structurelawsmultiplyassoc_secondresultbound + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplyassoc_secondresultcongruence pfa_offset_right_finite_structurelawsmultiplyassoc_secondresultcongruence. ((pfa_law_b_finite_structurelaws) * (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplyassoc_secondresultcongruence = (pfa_law_y_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplyassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsmultiplyassoc_rightleft. pfa_gap_finite_structurelawsmultiplyassoc_rightleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiplyassoc_rightright. pfa_gap_finite_structurelawsmultiplyassoc_rightright + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiplyassoc_rightresultbound. pfa_gap_finite_structurelawsmultiplyassoc_rightresultbound + S (pfa_law_v_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiplyassoc_rightresultcongruence pfa_offset_right_finite_structurelawsmultiplyassoc_rightresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_y_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiplyassoc_rightresultcongruence = (pfa_law_v_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiplyassoc_rightresultcongruence))))))))) -> pfa_law_u_finite_structurelaws = pfa_law_v_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws pfa_law_s_finite_structurelaws pfa_law_x_finite_structurelaws pfa_law_y_finite_structurelaws pfa_law_u_finite_structurelaws pfa_law_v_finite_structurelaws. (((exists pfa_gap_finite_structurelawsleftdistribution_sumleft. pfa_gap_finite_structurelawsleftdistribution_sumleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsleftdistribution_sumright. pfa_gap_finite_structurelawsleftdistribution_sumright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsleftdistribution_sumresultbound. pfa_gap_finite_structurelawsleftdistribution_sumresultbound + S (pfa_law_s_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsleftdistribution_sumresultcongruence pfa_offset_right_finite_structurelawsleftdistribution_sumresultcongruence. ((pfa_law_b_finite_structurelaws) + (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsleftdistribution_sumresultcongruence = (pfa_law_s_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsleftdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsleftdistribution_leftleft. pfa_gap_finite_structurelawsleftdistribution_leftleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsleftdistribution_leftright. pfa_gap_finite_structurelawsleftdistribution_leftright + S (pfa_law_s_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsleftdistribution_leftresultbound. pfa_gap_finite_structurelawsleftdistribution_leftresultbound + S (pfa_law_u_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsleftdistribution_leftresultcongruence pfa_offset_right_finite_structurelawsleftdistribution_leftresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_s_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsleftdistribution_leftresultcongruence = (pfa_law_u_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsleftdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsleftdistribution_firstleft. pfa_gap_finite_structurelawsleftdistribution_firstleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsleftdistribution_firstright. pfa_gap_finite_structurelawsleftdistribution_firstright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsleftdistribution_firstresultbound. pfa_gap_finite_structurelawsleftdistribution_firstresultbound + S (pfa_law_x_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsleftdistribution_firstresultcongruence pfa_offset_right_finite_structurelawsleftdistribution_firstresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsleftdistribution_firstresultcongruence = (pfa_law_x_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsleftdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsleftdistribution_secondleft. pfa_gap_finite_structurelawsleftdistribution_secondleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsleftdistribution_secondright. pfa_gap_finite_structurelawsleftdistribution_secondright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsleftdistribution_secondresultbound. pfa_gap_finite_structurelawsleftdistribution_secondresultbound + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsleftdistribution_secondresultcongruence pfa_offset_right_finite_structurelawsleftdistribution_secondresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsleftdistribution_secondresultcongruence = (pfa_law_y_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsleftdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsleftdistribution_rightleft. pfa_gap_finite_structurelawsleftdistribution_rightleft + S (pfa_law_x_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsleftdistribution_rightright. pfa_gap_finite_structurelawsleftdistribution_rightright + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsleftdistribution_rightresultbound. pfa_gap_finite_structurelawsleftdistribution_rightresultbound + S (pfa_law_v_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsleftdistribution_rightresultcongruence pfa_offset_right_finite_structurelawsleftdistribution_rightresultcongruence. ((pfa_law_x_finite_structurelaws) + (pfa_law_y_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsleftdistribution_rightresultcongruence = (pfa_law_v_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsleftdistribution_rightresultcongruence))))))))) -> pfa_law_u_finite_structurelaws = pfa_law_v_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws pfa_law_c_finite_structurelaws pfa_law_s_finite_structurelaws pfa_law_x_finite_structurelaws pfa_law_y_finite_structurelaws pfa_law_u_finite_structurelaws pfa_law_v_finite_structurelaws. (((exists pfa_gap_finite_structurelawsrightdistribution_sumleft. pfa_gap_finite_structurelawsrightdistribution_sumleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsrightdistribution_sumright. pfa_gap_finite_structurelawsrightdistribution_sumright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsrightdistribution_sumresultbound. pfa_gap_finite_structurelawsrightdistribution_sumresultbound + S (pfa_law_s_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsrightdistribution_sumresultcongruence pfa_offset_right_finite_structurelawsrightdistribution_sumresultcongruence. ((pfa_law_b_finite_structurelaws) + (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsrightdistribution_sumresultcongruence = (pfa_law_s_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsrightdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsrightdistribution_leftleft. pfa_gap_finite_structurelawsrightdistribution_leftleft + S (pfa_law_s_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsrightdistribution_leftright. pfa_gap_finite_structurelawsrightdistribution_leftright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsrightdistribution_leftresultbound. pfa_gap_finite_structurelawsrightdistribution_leftresultbound + S (pfa_law_u_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsrightdistribution_leftresultcongruence pfa_offset_right_finite_structurelawsrightdistribution_leftresultcongruence. ((pfa_law_s_finite_structurelaws) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsrightdistribution_leftresultcongruence = (pfa_law_u_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsrightdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsrightdistribution_firstleft. pfa_gap_finite_structurelawsrightdistribution_firstleft + S (pfa_law_b_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsrightdistribution_firstright. pfa_gap_finite_structurelawsrightdistribution_firstright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsrightdistribution_firstresultbound. pfa_gap_finite_structurelawsrightdistribution_firstresultbound + S (pfa_law_x_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsrightdistribution_firstresultcongruence pfa_offset_right_finite_structurelawsrightdistribution_firstresultcongruence. ((pfa_law_b_finite_structurelaws) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsrightdistribution_firstresultcongruence = (pfa_law_x_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsrightdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsrightdistribution_secondleft. pfa_gap_finite_structurelawsrightdistribution_secondleft + S (pfa_law_c_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsrightdistribution_secondright. pfa_gap_finite_structurelawsrightdistribution_secondright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsrightdistribution_secondresultbound. pfa_gap_finite_structurelawsrightdistribution_secondresultbound + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsrightdistribution_secondresultcongruence pfa_offset_right_finite_structurelawsrightdistribution_secondresultcongruence. ((pfa_law_c_finite_structurelaws) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsrightdistribution_secondresultcongruence = (pfa_law_y_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsrightdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_finite_structurelawsrightdistribution_rightleft. pfa_gap_finite_structurelawsrightdistribution_rightleft + S (pfa_law_x_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsrightdistribution_rightright. pfa_gap_finite_structurelawsrightdistribution_rightright + S (pfa_law_y_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsrightdistribution_rightresultbound. pfa_gap_finite_structurelawsrightdistribution_rightresultbound + S (pfa_law_v_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsrightdistribution_rightresultcongruence pfa_offset_right_finite_structurelawsrightdistribution_rightresultcongruence. ((pfa_law_x_finite_structurelaws) + (pfa_law_y_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsrightdistribution_rightresultcongruence = (pfa_law_v_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsrightdistribution_rightresultcongruence))))))))) -> pfa_law_u_finite_structurelaws = pfa_law_v_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsadd_zero_rightinput. pfa_gap_finite_structurelawsadd_zero_rightinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsadd_zero_rightleft. pfa_gap_finite_structurelawsadd_zero_rightleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsadd_zero_rightright. pfa_gap_finite_structurelawsadd_zero_rightright + S (0) = (p)) /\ ((((exists pfa_gap_finite_structurelawsadd_zero_rightresultbound. pfa_gap_finite_structurelawsadd_zero_rightresultbound + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsadd_zero_rightresultcongruence pfa_offset_right_finite_structurelawsadd_zero_rightresultcongruence. ((pfa_law_a_finite_structurelaws) + (0)) + (p) * pfa_offset_left_finite_structurelawsadd_zero_rightresultcongruence = (pfa_law_a_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsadd_zero_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsadd_zero_leftinput. pfa_gap_finite_structurelawsadd_zero_leftinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsadd_zero_leftleft. pfa_gap_finite_structurelawsadd_zero_leftleft + S (0) = (p)) /\ (((exists pfa_gap_finite_structurelawsadd_zero_leftright. pfa_gap_finite_structurelawsadd_zero_leftright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsadd_zero_leftresultbound. pfa_gap_finite_structurelawsadd_zero_leftresultbound + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsadd_zero_leftresultcongruence pfa_offset_right_finite_structurelawsadd_zero_leftresultcongruence. ((0) + (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsadd_zero_leftresultcongruence = (pfa_law_a_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsadd_zero_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsmultiply_one_rightinput. pfa_gap_finite_structurelawsmultiply_one_rightinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsmultiply_one_rightleft. pfa_gap_finite_structurelawsmultiply_one_rightleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiply_one_rightright. pfa_gap_finite_structurelawsmultiply_one_rightright + S (1) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiply_one_rightresultbound. pfa_gap_finite_structurelawsmultiply_one_rightresultbound + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiply_one_rightresultcongruence pfa_offset_right_finite_structurelawsmultiply_one_rightresultcongruence. ((pfa_law_a_finite_structurelaws) * (1)) + (p) * pfa_offset_left_finite_structurelawsmultiply_one_rightresultcongruence = (pfa_law_a_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiply_one_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsmultiply_one_leftinput. pfa_gap_finite_structurelawsmultiply_one_leftinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsmultiply_one_leftleft. pfa_gap_finite_structurelawsmultiply_one_leftleft + S (1) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiply_one_leftright. pfa_gap_finite_structurelawsmultiply_one_leftright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiply_one_leftresultbound. pfa_gap_finite_structurelawsmultiply_one_leftresultbound + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiply_one_leftresultcongruence pfa_offset_right_finite_structurelawsmultiply_one_leftresultcongruence. ((1) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiply_one_leftresultcongruence = (pfa_law_a_finite_structurelaws) + (p) * pfa_offset_right_finite_structurelawsmultiply_one_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsmultiply_zero_rightinput. pfa_gap_finite_structurelawsmultiply_zero_rightinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsmultiply_zero_rightleft. pfa_gap_finite_structurelawsmultiply_zero_rightleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiply_zero_rightright. pfa_gap_finite_structurelawsmultiply_zero_rightright + S (0) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiply_zero_rightresultbound. pfa_gap_finite_structurelawsmultiply_zero_rightresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiply_zero_rightresultcongruence pfa_offset_right_finite_structurelawsmultiply_zero_rightresultcongruence. ((pfa_law_a_finite_structurelaws) * (0)) + (p) * pfa_offset_left_finite_structurelawsmultiply_zero_rightresultcongruence = (0) + (p) * pfa_offset_right_finite_structurelawsmultiply_zero_rightresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsmultiply_zero_leftinput. pfa_gap_finite_structurelawsmultiply_zero_leftinput + S (pfa_law_a_finite_structurelaws) = (p)) -> (((exists pfa_gap_finite_structurelawsmultiply_zero_leftleft. pfa_gap_finite_structurelawsmultiply_zero_leftleft + S (0) = (p)) /\ (((exists pfa_gap_finite_structurelawsmultiply_zero_leftright. pfa_gap_finite_structurelawsmultiply_zero_leftright + S (pfa_law_a_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsmultiply_zero_leftresultbound. pfa_gap_finite_structurelawsmultiply_zero_leftresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsmultiply_zero_leftresultcongruence pfa_offset_right_finite_structurelawsmultiply_zero_leftresultcongruence. ((0) * (pfa_law_a_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsmultiply_zero_leftresultcongruence = (0) + (p) * pfa_offset_right_finite_structurelawsmultiply_zero_leftresultcongruence)))))))))) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsnegateinput. pfa_gap_finite_structurelawsnegateinput + S (pfa_law_a_finite_structurelaws) = (p)) -> exists pfa_law_b_finite_structurelaws. (((exists pfa_gap_finite_structurelawsnegatechosenadditionleft. pfa_gap_finite_structurelawsnegatechosenadditionleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsnegatechosenadditionright. pfa_gap_finite_structurelawsnegatechosenadditionright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsnegatechosenadditionresultbound. pfa_gap_finite_structurelawsnegatechosenadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsnegatechosenadditionresultcongruence pfa_offset_right_finite_structurelawsnegatechosenadditionresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsnegatechosenadditionresultcongruence = (0) + (p) * pfa_offset_right_finite_structurelawsnegatechosenadditionresultcongruence))))))))) /\ forall pfa_law_c_finite_structurelaws. (((exists pfa_gap_finite_structurelawsnegateotheradditionleft. pfa_gap_finite_structurelawsnegateotheradditionleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsnegateotheradditionright. pfa_gap_finite_structurelawsnegateotheradditionright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsnegateotheradditionresultbound. pfa_gap_finite_structurelawsnegateotheradditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsnegateotheradditionresultcongruence pfa_offset_right_finite_structurelawsnegateotheradditionresultcongruence. ((pfa_law_a_finite_structurelaws) + (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsnegateotheradditionresultcongruence = (0) + (p) * pfa_offset_right_finite_structurelawsnegateotheradditionresultcongruence))))))))) -> pfa_law_c_finite_structurelaws = pfa_law_b_finite_structurelaws) /\ (((forall pfa_law_a_finite_structurelaws. (exists pfa_gap_finite_structurelawsinverseinput. pfa_gap_finite_structurelawsinverseinput + S (pfa_law_a_finite_structurelaws) = (p)) -> ~(pfa_law_a_finite_structurelaws = 0) -> exists pfa_law_b_finite_structurelaws. (((~((pfa_law_a_finite_structurelaws) = 0)) /\ ((((exists pfa_gap_finite_structurelawsinversechosenmultiplicationleft. pfa_gap_finite_structurelawsinversechosenmultiplicationleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsinversechosenmultiplicationright. pfa_gap_finite_structurelawsinversechosenmultiplicationright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsinversechosenmultiplicationresultbound. pfa_gap_finite_structurelawsinversechosenmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsinversechosenmultiplicationresultcongruence pfa_offset_right_finite_structurelawsinversechosenmultiplicationresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsinversechosenmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_finite_structurelawsinversechosenmultiplicationresultcongruence)))))))))))) /\ forall pfa_law_c_finite_structurelaws. (((~((pfa_law_a_finite_structurelaws) = 0)) /\ ((((exists pfa_gap_finite_structurelawsinverseothermultiplicationleft. pfa_gap_finite_structurelawsinverseothermultiplicationleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsinverseothermultiplicationright. pfa_gap_finite_structurelawsinverseothermultiplicationright + S (pfa_law_c_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsinverseothermultiplicationresultbound. pfa_gap_finite_structurelawsinverseothermultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsinverseothermultiplicationresultcongruence pfa_offset_right_finite_structurelawsinverseothermultiplicationresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_c_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsinverseothermultiplicationresultcongruence = (1) + (p) * pfa_offset_right_finite_structurelawsinverseothermultiplicationresultcongruence)))))))))))) -> pfa_law_c_finite_structurelaws = pfa_law_b_finite_structurelaws) /\ ((forall pfa_law_a_finite_structurelaws pfa_law_b_finite_structurelaws. (((exists pfa_gap_finite_structurelawsnozeroleft. pfa_gap_finite_structurelawsnozeroleft + S (pfa_law_a_finite_structurelaws) = (p)) /\ (((exists pfa_gap_finite_structurelawsnozeroright. pfa_gap_finite_structurelawsnozeroright + S (pfa_law_b_finite_structurelaws) = (p)) /\ ((((exists pfa_gap_finite_structurelawsnozeroresultbound. pfa_gap_finite_structurelawsnozeroresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_finite_structurelawsnozeroresultcongruence pfa_offset_right_finite_structurelawsnozeroresultcongruence. ((pfa_law_a_finite_structurelaws) * (pfa_law_b_finite_structurelaws)) + (p) * pfa_offset_left_finite_structurelawsnozeroresultcongruence = (0) + (p) * pfa_offset_right_finite_structurelawsnozeroresultcongruence))))))))) -> pfa_law_a_finite_structurelaws = 0 \/ pfa_law_b_finite_structurelaws = 0)))))))))))))))))))))))))))))))))))))))) /\ ((((exists pff_history_code_finite_structurecharacteristicmodulus pff_history_scale_finite_structurecharacteristicmodulus. (((((exists ff_h_pft_finite_structurecharacteristicmodulushistorystart. ff_h_pft_finite_structurecharacteristicmodulushistorystart + S (0) = S ((S (0)) * pff_history_scale_finite_structurecharacteristicmodulus)) /\ exists ff_q_pft_finite_structurecharacteristicmodulushistorystart. pff_history_code_finite_structurecharacteristicmodulus = ff_q_pft_finite_structurecharacteristicmodulushistorystart * S ((S (0)) * pff_history_scale_finite_structurecharacteristicmodulus) + (0))) /\ (((((exists ff_h_pft_finite_structurecharacteristicmodulushistoryterminal. ff_h_pft_finite_structurecharacteristicmodulushistoryterminal + S (0) = S ((S (p)) * pff_history_scale_finite_structurecharacteristicmodulus)) /\ exists ff_q_pft_finite_structurecharacteristicmodulushistoryterminal. pff_history_code_finite_structurecharacteristicmodulus = ff_q_pft_finite_structurecharacteristicmodulushistoryterminal * S ((S (p)) * pff_history_scale_finite_structurecharacteristicmodulus) + (0))) /\ ((forall pff_trace_index_finite_structurecharacteristicmodulushistorysteps. (exists pfa_gap_finite_structurecharacteristicmodulushistorystepsindex. pfa_gap_finite_structurecharacteristicmodulushistorystepsindex + S (pff_trace_index_finite_structurecharacteristicmodulushistorysteps) = (p)) -> exists pff_trace_before_finite_structurecharacteristicmodulushistorysteps pff_trace_after_finite_structurecharacteristicmodulushistorysteps. ((((exists ff_h_pft_finite_structurecharacteristicmodulushistorystepsbefore. ff_h_pft_finite_structurecharacteristicmodulushistorystepsbefore + S (pff_trace_before_finite_structurecharacteristicmodulushistorysteps) = S ((S (pff_trace_index_finite_structurecharacteristicmodulushistorysteps)) * pff_history_scale_finite_structurecharacteristicmodulus)) /\ exists ff_q_pft_finite_structurecharacteristicmodulushistorystepsbefore. pff_history_code_finite_structurecharacteristicmodulus = ff_q_pft_finite_structurecharacteristicmodulushistorystepsbefore * S ((S (pff_trace_index_finite_structurecharacteristicmodulushistorysteps)) * pff_history_scale_finite_structurecharacteristicmodulus) + (pff_trace_before_finite_structurecharacteristicmodulushistorysteps))) /\ (((((exists ff_h_pft_finite_structurecharacteristicmodulushistorystepsafter. ff_h_pft_finite_structurecharacteristicmodulushistorystepsafter + S (pff_trace_after_finite_structurecharacteristicmodulushistorysteps) = S ((S (S (pff_trace_index_finite_structurecharacteristicmodulushistorysteps))) * pff_history_scale_finite_structurecharacteristicmodulus)) /\ exists ff_q_pft_finite_structurecharacteristicmodulushistorystepsafter. pff_history_code_finite_structurecharacteristicmodulus = ff_q_pft_finite_structurecharacteristicmodulushistorystepsafter * S ((S (S (pff_trace_index_finite_structurecharacteristicmodulushistorysteps))) * pff_history_scale_finite_structurecharacteristicmodulus) + (pff_trace_after_finite_structurecharacteristicmodulushistorysteps))) /\ ((((exists pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionleft. pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionleft + S (pff_trace_before_finite_structurecharacteristicmodulushistorysteps) = (p)) /\ (((exists pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionright. pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionresultbound. pfa_gap_finite_structurecharacteristicmodulushistorystepsadditionresultbound + S (pff_trace_after_finite_structurecharacteristicmodulushistorysteps) = (p)) /\ ((exists pfa_offset_left_finite_structurecharacteristicmodulushistorystepsadditionresultcongruence pfa_offset_right_finite_structurecharacteristicmodulushistorystepsadditionresultcongruence. ((pff_trace_before_finite_structurecharacteristicmodulushistorysteps) + (1)) + (p) * pfa_offset_left_finite_structurecharacteristicmodulushistorystepsadditionresultcongruence = (pff_trace_after_finite_structurecharacteristicmodulushistorysteps) + (p) * pfa_offset_right_finite_structurecharacteristicmodulushistorystepsadditionresultcongruence)))))))))))))))))))) /\ ((forall pff_smaller_positive_finite_structurecharacteristic. (exists pfa_gap_finite_structurecharacteristicstrict. pfa_gap_finite_structurecharacteristicstrict + S (pff_smaller_positive_finite_structurecharacteristic) = (p)) -> ~(pff_smaller_positive_finite_structurecharacteristic = 0) -> ~(exists pff_history_code_finite_structurecharacteristicsmaller pff_history_scale_finite_structurecharacteristicsmaller. (((((exists ff_h_pft_finite_structurecharacteristicsmallerhistorystart. ff_h_pft_finite_structurecharacteristicsmallerhistorystart + S (0) = S ((S (0)) * pff_history_scale_finite_structurecharacteristicsmaller)) /\ exists ff_q_pft_finite_structurecharacteristicsmallerhistorystart. pff_history_code_finite_structurecharacteristicsmaller = ff_q_pft_finite_structurecharacteristicsmallerhistorystart * S ((S (0)) * pff_history_scale_finite_structurecharacteristicsmaller) + (0))) /\ (((((exists ff_h_pft_finite_structurecharacteristicsmallerhistoryterminal. ff_h_pft_finite_structurecharacteristicsmallerhistoryterminal + S (0) = S ((S (pff_smaller_positive_finite_structurecharacteristic)) * pff_history_scale_finite_structurecharacteristicsmaller)) /\ exists ff_q_pft_finite_structurecharacteristicsmallerhistoryterminal. pff_history_code_finite_structurecharacteristicsmaller = ff_q_pft_finite_structurecharacteristicsmallerhistoryterminal * S ((S (pff_smaller_positive_finite_structurecharacteristic)) * pff_history_scale_finite_structurecharacteristicsmaller) + (0))) /\ ((forall pff_trace_index_finite_structurecharacteristicsmallerhistorysteps. (exists pfa_gap_finite_structurecharacteristicsmallerhistorystepsindex. pfa_gap_finite_structurecharacteristicsmallerhistorystepsindex + S (pff_trace_index_finite_structurecharacteristicsmallerhistorysteps) = (pff_smaller_positive_finite_structurecharacteristic)) -> exists pff_trace_before_finite_structurecharacteristicsmallerhistorysteps pff_trace_after_finite_structurecharacteristicsmallerhistorysteps. ((((exists ff_h_pft_finite_structurecharacteristicsmallerhistorystepsbefore. ff_h_pft_finite_structurecharacteristicsmallerhistorystepsbefore + S (pff_trace_before_finite_structurecharacteristicsmallerhistorysteps) = S ((S (pff_trace_index_finite_structurecharacteristicsmallerhistorysteps)) * pff_history_scale_finite_structurecharacteristicsmaller)) /\ exists ff_q_pft_finite_structurecharacteristicsmallerhistorystepsbefore. pff_history_code_finite_structurecharacteristicsmaller = ff_q_pft_finite_structurecharacteristicsmallerhistorystepsbefore * S ((S (pff_trace_index_finite_structurecharacteristicsmallerhistorysteps)) * pff_history_scale_finite_structurecharacteristicsmaller) + (pff_trace_before_finite_structurecharacteristicsmallerhistorysteps))) /\ (((((exists ff_h_pft_finite_structurecharacteristicsmallerhistorystepsafter. ff_h_pft_finite_structurecharacteristicsmallerhistorystepsafter + S (pff_trace_after_finite_structurecharacteristicsmallerhistorysteps) = S ((S (S (pff_trace_index_finite_structurecharacteristicsmallerhistorysteps))) * pff_history_scale_finite_structurecharacteristicsmaller)) /\ exists ff_q_pft_finite_structurecharacteristicsmallerhistorystepsafter. pff_history_code_finite_structurecharacteristicsmaller = ff_q_pft_finite_structurecharacteristicsmallerhistorystepsafter * S ((S (S (pff_trace_index_finite_structurecharacteristicsmallerhistorysteps))) * pff_history_scale_finite_structurecharacteristicsmaller) + (pff_trace_after_finite_structurecharacteristicsmallerhistorysteps))) /\ ((((exists pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionleft. pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionleft + S (pff_trace_before_finite_structurecharacteristicsmallerhistorysteps) = (p)) /\ (((exists pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionright. pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionresultbound. pfa_gap_finite_structurecharacteristicsmallerhistorystepsadditionresultbound + S (pff_trace_after_finite_structurecharacteristicsmallerhistorysteps) = (p)) /\ ((exists pfa_offset_left_finite_structurecharacteristicsmallerhistorystepsadditionresultcongruence pfa_offset_right_finite_structurecharacteristicsmallerhistorystepsadditionresultcongruence. ((pff_trace_before_finite_structurecharacteristicsmallerhistorysteps) + (1)) + (p) * pfa_offset_left_finite_structurecharacteristicsmallerhistorystepsadditionresultcongruence = (pff_trace_after_finite_structurecharacteristicsmallerhistorysteps) + (p) * pfa_offset_right_finite_structurecharacteristicsmallerhistorystepsadditionresultcongruence)))))))))))))))))))))))))))))))

forall n m k. (n + m) + k = n + (m + k)

forall n m. n + m = m + n

forall n m k. (n + m) * k = n * k + m * k

forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

forall b c i x y. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y

forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists q. z = q * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. z = q * S ((S i) * c) + a))

forall m n r s. (((exists ff_gap_binary_value. ff_gap_binary_value + S (r) = m) /\ (exists ff_left_binary_value_congruence ff_right_binary_value_congruence. (n) + m * ff_left_binary_value_congruence = (r) + m * ff_right_binary_value_congruence))) -> (((exists ff_gap_binary_other. ff_gap_binary_other + S (s) = m) /\ (exists ff_left_binary_other_congruence ff_right_binary_other_congruence. (n) + m * ff_left_binary_other_congruence = (s) + m * ff_right_binary_other_congruence))) -> r = s

forall p x y z. (exists wip_strict_gap_unique_y_bound. wip_strict_gap_unique_y_bound + S y = p) -> (exists wip_strict_gap_unique_z_bound. wip_strict_gap_unique_z_bound + S z = p) -> (exists wip_mod_left_unique_xy wip_mod_right_unique_xy. x * y + p * wip_mod_left_unique_xy = 1 + p * wip_mod_right_unique_xy) -> (exists wip_mod_left_unique_xz wip_mod_right_unique_xz. x * z + p * wip_mod_left_unique_xz = 1 + p * wip_mod_right_unique_xz) -> y = z

forall m n. ~(m = 0) -> exists q r. n = m * q + r /\ S r <= m

forall m n q r q2 r2. n = m * q + r -> (exists k. k + S r = m) -> n = m * q2 + r2 -> (exists k. k + S r2 = m) -> q = q2 /\ r = r2

forall a b. a = b \/ ~(a = b)

forall n x. (exists h. h + S x = S n) -> x = n \/ exists h. h + S x = n

forall m a. ~(m = 0) -> exists r. ((exists hpl_gap_bound. hpl_gap_bound + S (r) = (m)) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. a + m * hgcrt_mod_left_hpl_mod = r + m * hgcrt_mod_right_hpl_mod))

forall a b. (exists k. k + a = b) -> exists r. r + a = S b

forall a b. (exists k. k + S a = b) -> ~ (exists k. k + b = a)

forall d. exists b c. (forall mdr_i_identity_exists. (exists mdr_gap_identity_existsbound. mdr_gap_identity_existsbound + S (mdr_i_identity_exists) = (d)) -> (((exists ff_h_mdr_identity_existsentry. ff_h_mdr_identity_existsentry + S (mdr_i_identity_exists) = S ((S (mdr_i_identity_exists)) * c)) /\ exists ff_q_mdr_identity_existsentry. b = ff_q_mdr_identity_existsentry * S ((S (mdr_i_identity_exists)) * c) + (mdr_i_identity_exists))))

forall w r s. (exists mdr_gap_flat_row. mdr_gap_flat_row + S (r) = (w)) -> (exists mdr_gap_flat_column. mdr_gap_flat_column + S (s) = (w)) -> (exists mdr_gap_flat_result. mdr_gap_flat_result + S (r * w + s) = (w * w))

forall q k r s. k = q * r + s -> (exists mdr_gap_quotient_source. mdr_gap_quotient_source + S (k) = (q * q)) -> (exists mdr_gap_quotient_row. mdr_gap_quotient_row + S (r) = (q))

forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a + c) + m * x = (b + d) + m * y

forall d a b c. (exists fspm_u_cancel_input fspm_v_cancel_input. (a + b) + d * fspm_u_cancel_input = (a + c) + d * fspm_v_cancel_input) -> (exists fspm_u_cancel_result fspm_v_cancel_result. (b) + d * fspm_u_cancel_result = (c) + d * fspm_v_cancel_result)

forall m a b. (exists ha. ha + S a = m) -> (exists hb. hb + S b = m) -> (exists u v. a + m * u = b + m * v) -> a = b

forall m a x y. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> (exists u v. (a * x) + m * u = (a * y) + m * v) -> exists r s. x + m * r = y + m * s

forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a * c) + m * x = (b * d) + m * y

forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (c * a) + m * r = (c * b) + m * s

forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (a * c) + m * r = (b * c) + m * s

forall m a. exists u v. a + m * u = a + m * v

forall m a b. (exists u v. a + m * u = b + m * v) -> exists r s. b + m * r = a + m * s

forall m a b c. (exists u v. a + m * u = b + m * v) -> (exists r s. b + m * r = c + m * s) -> exists x y. a + m * x = c + m * y

forall a m z. (exists hmi_left_offset_converse_assumption hmi_right_offset_converse_assumption. a * z + m * hmi_left_offset_converse_assumption = 1 + m * hmi_right_offset_converse_assumption) -> (forall hmi_divisor_converse_result. (exists hmi_left_factor_converse_result. a = hmi_divisor_converse_result * hmi_left_factor_converse_result) -> (exists hmi_right_factor_converse_result. m = hmi_divisor_converse_result * hmi_right_factor_converse_result) -> hmi_divisor_converse_result = 1)

forall n m k. n * (m + k) = n * m + n * k

forall n m k. (n * m) * k = n * (m * k)

forall n m. n * m = m * n

forall n. n * 1 = n

forall n. ~(n = 0) -> 1 <= n

forall p a. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> ~(a = 0) -> (exists qrbu_gap_a_lt_p. qrbu_gap_a_lt_p + S a = p) -> (exists qrbu_inverse_bounded_inverse. (~(qrbu_inverse_bounded_inverse = 0) /\ ((exists qrbu_gap_bounded_inverse_bound. qrbu_gap_bounded_inverse_bound + S qrbu_inverse_bounded_inverse = p) /\ (exists qrbu_mod_left_bounded_inverse_mod qrbu_mod_right_bounded_inverse_mod. a * qrbu_inverse_bounded_inverse + p * qrbu_mod_left_bounded_inverse_mod = 1 + p * qrbu_mod_right_bounded_inverse_mod))))

forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)

forall p. ((~(p = 1) /\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \/ frm_prime_right_bpvl_prime = 1)) -> (exists bpvl_gap_prime_two. bpvl_gap_prime_two + (2) = (p))

forall xp xn m. ~(m = 0) -> exists qp qn r. (((xp) + (m) * (qn) = ((xn) + (m) * (qp)) + (r) /\ exists sif_gap_pair_total. sif_gap_pair_total + S (r) = (m)))

forall a b. (exists k. k + a = b) -> exists r. r + S a = S b

forall n. ~(S n = 0)

forall n. 0 + n = n

forall n. 0 <= n