% % 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. % % Current problem (Lemma 3.1) % formulas(assumptions). % multi-part lemma negated by hand (A ^ B) -> B != 1 % (3.10), Lemma 3.1 (1) | (A ^ B) => B != 1 % (3.11), Lemma 3.1 (2) | ~A => ~B != B => A % (3.12), Lemma 3.1 (3) | (A => B) ^ B != B % (3.13), Lemma 3.1 (4) | A => (A ^ B) != A => B % (3.14), Lemma 3.1 (5) # answer("(Lemma 3.1, 5 parts)"). end_of_list. % % Hints extracted from previously found proofs. % formulas(hints). (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 -> ~ y) ^ (y -> ~ x) = x => ~ y. 1 ^ (x -> y) = x -> (x ^ y). (x => y) ^ z = (x -> y) ^ ((~ y -> ~ x) ^ z). (x -> y) ^ ((~ y -> ~ x) ^ z) = (x => y) ^ z. x ^ (y v ~ x) = x ^ (x -> y). (~ x -> ~ y) ^ (y -> x) = y => x. x v (y ^ x) = x. x ^ (y v x) = x. (x -> y) ^ (x -> 1) = x -> y. x = 1 ^ x. 1 ^ x = x. 1 ^ (x -> y) = (y v x) -> y. x ^ (y -> x) = (1 v y) -> x. 1 v x = 1. x ^ (y v (z v x)) = x. x ^ (y v (x v z)) = x. x -> (x ^ y) = x -> y. (x -> y) ^ (~ (x ^ y) -> ~ x) = x => (x ^ y). x -> x = x -> (x v y). x -> (x v y) = x -> x. x -> (x ^ (x -> y)) = x -> y. x -> (x v y) = 1. ~ x -> ~ (x ^ y) = 1. x -> (y v x) = 1. (x ^ y) -> y = 1. ((x ^ y) -> x) ^ 1 = (x ^ y) => x. (x v y) -> x = y -> x. ~ (x ^ y) -> ~ x = ~ y -> ~ x. x ^ (y -> x) = 1 -> x. x ^ (y -> x) = x. x = (y -> x) ^ x. (x -> y) ^ y = y. (x -> y) v y = x -> y. x -> (y -> x) = 1. (x -> (y -> x)) ^ (x -> 1) = 1. x ^ (y -> (x v z)) = x. (x -> (y -> x)) ^ 1 = 1. x -> (x ^ (x -> y)) = x -> (y v ~ x). (x -> ~ y) ^ (y -> ~ x) = y => ~ x. (x -> (y -> x)) ^ 1 = (x ^ y) => x. (x ^ y) => x = 1. (x ^ y) => y = 1. x -> (y v ~ x) = x -> y. x -> (~ x v y) = x -> y. ~ x ^ (x -> y) = ~ x. x ^ (~ x -> y) = ~ ~ x. x ^ (~ x -> y) = x. x = (~ x -> y) ^ x. (~ x -> y) ^ x = x. x => ~ y = y => ~ x. x => y = ~ y => ~ x. ~ x => ~ y = y => x. (x -> y) ^ y = (x => y) ^ y. (x => y) ^ y = y. (x -> y) ^ (~ y -> ~ x) = x => (x ^ y). x => (x ^ y) = x => y. end_of_list.