% % Prover9 commands % clear(auto). op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). lex([0,1,*,^,=>,->,v,~,-]). assign(order,kbo). set(lex_order_vars). set(paramodulation). set(para_units_only). clear(back_demod). set(hyper_resolution). set(auto_denials). set(restrict_denials). % ************************************************************************** % Clauses for Section 3 % ************************************************************************** formulas(assumptions). % % Identities axiomatising bounded distributive lattices % (x v y) v z = x v (y v z) # label("(D1)"). (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). x v y = y v x # label("(D3)"). x ^ y = y ^ x # label("(D4)"). x v (x ^ y) = x # label("(D5)"). x ^ (x v y) = x # label("(D6)"). (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). x v 0 = x # label("(D9)"). x ^ 1 = x # label("(D10)"). % % Identities axiomatising De Morgan algebras % % Note: These identities axiomatise De Morgan algebras when taken in % conjunction with the identities axiomatising bounded distributive % lattices given above. % x ^ 0 = 0 # label("(2.1)"). ~x v ~y = ~(x ^ y) # label("(DM1)"). ~x ^ ~y = ~(x v y) # label("(DM2)"). ~(~x) = x # label("(DN)"). ~1 = 0 # label("(2.2)"). % % Identities axiomatising Nelson algebras % % Note: These identities axiomatise Nelson algebras when taken in % conjunction with the identities axiomatising De Morgan algebras % given above. % (x ^ ~x) ^ (y v ~y) = x ^ ~x # label("(N1)"). x -> x = 1 # label("(N2)"). (x -> y) ^ (~x v y) = ~x v y # label("(N3)"). x ^ (~x v y) = x ^ (x -> y) # label("(N4)"). (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). (x ^ y) -> z = x -> (y -> z) # label("(N6)"). -x = x -> 0 # label("(N7)"). % % Definitions mapping Nelson algebras to Nelson FL_ew algebras % % Definition of * (monoid operation) in Nelson algebras % Note: (* def) is equivalent to x * y = ~(x => ~y). x * y = ~(x -> ~y) v ~(y -> ~x) # label("(* def)"). % Definition of => (strong implication) in Nelson algebras x => y = (x -> y) ^ (~y -> ~x) # label("(=> def)"). % % Identities known to be satisfied by the variety of Nelson algebras % x -> 1 = 1 # label("(3.1)"). 1 -> x = x # label("(3.2)"). (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). x -> (y -> z) = y -> (x -> z) # label("(3.4)"). x -> (x -> y) = x -> y # label("(3.5)"). (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). x ^ ((x => y) => y) = x # label("(3.7)"). x -> ~(x -> y) = x -> ~y # label("(3.8)"). ~(x -> y) -> z = x -> (~y -> z) # label("(3.9)"). % % BCK-algebra identities and quasi-identities known to be % satisfied by the variety of Nelson algebras % % Note: BCK1, BCK1', BCK5 are not included here. % 1 => x = x # label("(BCK2)"). x => 1 = 1 # label("(BCK3)"). x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). end_of_list. % % Previous results % formulas(assumptions). (x ^ y) -> y = 1 # label("(3.10)"). % Lemma 3.1 (1) (x ^ y) => y = 1 # label("(3.11)"). % Lemma 3.1 (2) ~x => ~y = y => x # label("(3.12)"). % Lemma 3.1 (3) (x => y) ^ y = y # label("(3.13)"). % Lemma 3.1 (4) x => (x ^ y) = x => y # label("(3.14)"). % Lemma 3.1 (5) x => (x => y) = x -> y # label("(3.15)"). % Prop 3.2 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). % Lemma 3.3 end_of_list. % % Current problem (Lemma 3.4) % formulas(goals). x => ((x => y) => y) = 1 # label("(BCK5)"). % Lemma 3.4 end_of_list. % % Hints extracted from previously found proofs. % formulas(hints). x => ((x => y) => y) = 1. end_of_list.