0.12/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.12/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.34 % Computer : n008.cluster.edu 0.12/0.34 % Model : x86_64 x86_64 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.34 % Memory : 8042.1875MB 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.34 % CPULimit : 180 0.12/0.34 % DateTime : Thu Aug 29 14:17:53 EDT 2019 0.12/0.34 % CPUTime : 22.08/22.31 % SZS status Unsatisfiable 22.08/22.31 22.39/22.56 % SZS output start Proof 22.39/22.56 Take the following subset of the input axioms: 22.62/22.78 fof(axBp, axiom, ![A, B, C]: p(i(i(A, B), i(i(B, C), i(A, C))))=true). 22.62/22.78 fof(axK_axC, negated_conjecture, ![A, B, C]: tuple(p(i(i(A, i(B, C)), i(B, i(A, C)))), p(i(c1, i(c2, c1))))!=tuple(true, true)). 22.62/22.78 fof(axL, axiom, ![X, Y]: p(i(i(i(X, Y), Y), i(i(Y, X), X)))=true). 22.62/22.78 fof(axTO, axiom, ![X, Y]: true=p(i(i(i(X, Y), i(Y, X)), i(Y, X)))). 22.62/22.78 fof(ifeq_axiom, axiom, ![A, B, C]: ifeq(A, A, B, C)=B). 22.62/22.78 fof(modus_ponens, axiom, ![A, B]: true=ifeq(p(i(A, B)), true, ifeq(p(A), true, p(B), true), true)). 22.62/22.78 fof(resid1, axiom, ![A, B, C]: true=ifeq(p(i(f(A, B), C)), true, p(i(A, i(B, C))), true)). 22.62/22.78 fof(resid2, axiom, ![A, B, C]: ifeq(p(i(A, i(B, C))), true, p(i(f(A, B), C)), true)=true). 22.62/22.78 22.62/22.78 Now clausify the problem and encode Horn clauses using encoding 3 of 22.62/22.78 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 22.62/22.78 We repeatedly replace C & s=t => u=v by the two clauses: 22.62/22.78 fresh(y, y, x1...xn) = u 22.62/22.78 C => fresh(s, t, x1...xn) = v 22.62/22.78 where fresh is a fresh function symbol and x1..xn are the free 22.62/22.78 variables of u and v. 22.62/22.78 A predicate p(X) is encoded as p(X)=true (this is sound, because the 22.62/22.78 input problem has no model of domain size 1). 22.62/22.78 22.62/22.78 The encoding turns the above axioms into the following unit equations and goals: 22.62/22.78 22.62/22.78 Axiom 1 (axBp): p(i(i(X, Y), i(i(Y, Z), i(X, Z)))) = true. 22.62/22.78 Axiom 2 (axL): p(i(i(i(X, Y), Y), i(i(Y, X), X))) = true. 22.62/22.78 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y. 22.62/22.78 Axiom 4 (modus_ponens): true = ifeq(p(i(X, Y)), true, ifeq(p(X), true, p(Y), true), true). 22.62/22.78 Axiom 5 (axTO): true = p(i(i(i(X, Y), i(Y, X)), i(Y, X))). 22.62/22.78 Axiom 6 (resid1): true = ifeq(p(i(f(X, Y), Z)), true, p(i(X, i(Y, Z))), true). 23.25/23.48 Axiom 7 (resid2): ifeq(p(i(X, i(Y, Z))), true, p(i(f(X, Y), Z)), true) = true. 23.25/23.48 23.25/23.48 Lemma 8: ifeq(p(i(X, Y)), true, p(i(i(Y, Z), i(X, Z))), true) = true. 23.25/23.48 Proof: 23.25/23.48 ifeq(p(i(X, Y)), true, p(i(i(Y, Z), i(X, Z))), true) 23.25/23.48 = { by axiom 3 (ifeq_axiom) } 23.25/23.48 ifeq(true, true, ifeq(p(i(X, Y)), true, p(i(i(Y, Z), i(X, Z))), true), true) 23.25/23.48 = { by axiom 1 (axBp) } 23.25/23.48 ifeq(p(i(i(X, Y), i(i(Y, Z), i(X, Z)))), true, ifeq(p(i(X, Y)), true, p(i(i(Y, Z), i(X, Z))), true), true) 23.25/23.48 = { by axiom 4 (modus_ponens) } 23.25/23.48 true 23.25/23.48 23.25/23.48 Lemma 9: p(i(i(i(i(X, Y), i(Z, Y)), W), i(i(Z, X), W))) = true. 23.25/23.48 Proof: 23.25/23.48 p(i(i(i(i(X, Y), i(Z, Y)), W), i(i(Z, X), W))) 23.25/23.48 = { by axiom 3 (ifeq_axiom) } 23.25/23.48 ifeq(true, true, p(i(i(i(i(X, Y), i(Z, Y)), W), i(i(Z, X), W))), true) 23.25/23.48 = { by axiom 1 (axBp) } 23.25/23.48 ifeq(p(i(i(Z, X), i(i(X, Y), i(Z, Y)))), true, p(i(i(i(i(X, Y), i(Z, Y)), W), i(i(Z, X), W))), true) 23.25/23.48 = { by lemma 8 } 23.25/23.48 true 23.25/23.48 23.25/23.48 Lemma 10: ifeq(p(i(i(i(i(X, Y), i(Y, X)), i(Y, X)), Z)), true, p(Z), true) = true. 23.25/23.48 Proof: 23.25/23.48 ifeq(p(i(i(i(i(X, Y), i(Y, X)), i(Y, X)), Z)), true, p(Z), true) 23.25/23.48 = { by axiom 3 (ifeq_axiom) } 23.25/23.48 ifeq(p(i(i(i(i(X, Y), i(Y, X)), i(Y, X)), Z)), true, ifeq(true, true, p(Z), true), true) 23.25/23.48 = { by axiom 5 (axTO) } 23.25/23.48 ifeq(p(i(i(i(i(X, Y), i(Y, X)), i(Y, X)), Z)), true, ifeq(p(i(i(i(X, Y), i(Y, X)), i(Y, X))), true, p(Z), true), true) 23.25/23.48 = { by axiom 4 (modus_ponens) } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 11: p(i(X, X)) = true. 23.25/23.49 Proof: 23.25/23.49 p(i(X, X)) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, p(i(X, X)), true) 23.25/23.49 = { by lemma 10 } 23.25/23.49 ifeq(ifeq(p(i(i(i(i(X, X), i(X, X)), i(X, X)), i(i(X, X), i(X, X)))), true, p(i(i(X, X), i(X, X))), true), true, p(i(X, X)), true) 23.25/23.49 = { by lemma 9 } 23.25/23.49 ifeq(ifeq(true, true, p(i(i(X, X), i(X, X))), true), true, p(i(X, X)), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(p(i(i(X, X), i(X, X))), true, p(i(X, X)), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, ifeq(p(i(i(X, X), i(X, X))), true, p(i(X, X)), true), true) 23.25/23.49 = { by axiom 5 (axTO) } 23.25/23.49 ifeq(p(i(i(i(X, X), i(X, X)), i(X, X))), true, ifeq(p(i(i(X, X), i(X, X))), true, p(i(X, X)), true), true) 23.25/23.49 = { by axiom 4 (modus_ponens) } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 12: p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))) = true. 23.25/23.49 Proof: 23.25/23.49 p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))), true) 23.25/23.49 = { by axiom 7 (resid2) } 23.25/23.49 ifeq(ifeq(p(i(i(X, W), i(i(W, Y), i(X, Y)))), true, p(i(f(i(X, W), i(W, Y)), i(X, Y))), true), true, p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))), true) 23.25/23.49 = { by axiom 1 (axBp) } 23.25/23.49 ifeq(ifeq(true, true, p(i(f(i(X, W), i(W, Y)), i(X, Y))), true), true, p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(p(i(f(i(X, W), i(W, Y)), i(X, Y))), true, p(i(i(i(X, Y), Z), i(f(i(X, W), i(W, Y)), Z))), true) 23.25/23.49 = { by lemma 8 } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 13: p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))) = true. 23.25/23.49 Proof: 23.25/23.49 p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))), true) 23.25/23.49 = { by lemma 10 } 23.25/23.49 ifeq(ifeq(p(i(i(i(i(X, Y), i(Y, X)), i(Y, X)), i(f(i(i(X, Y), Z), i(Z, i(Y, X))), i(Y, X)))), true, p(i(f(i(i(X, Y), Z), i(Z, i(Y, X))), i(Y, X))), true), true, p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))), true) 23.25/23.49 = { by lemma 12 } 23.25/23.49 ifeq(ifeq(true, true, p(i(f(i(i(X, Y), Z), i(Z, i(Y, X))), i(Y, X))), true), true, p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(p(i(f(i(i(X, Y), Z), i(Z, i(Y, X))), i(Y, X))), true, p(i(i(i(X, Y), Z), i(i(Z, i(Y, X)), i(Y, X)))), true) 23.25/23.49 = { by axiom 6 (resid1) } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 14: ifeq(p(i(i(i(X, Y), i(Z, Y)), W)), true, p(i(i(Z, X), W)), true) = true. 23.25/23.49 Proof: 23.25/23.49 ifeq(p(i(i(i(X, Y), i(Z, Y)), W)), true, p(i(i(Z, X), W)), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, ifeq(p(i(i(i(X, Y), i(Z, Y)), W)), true, p(i(i(Z, X), W)), true), true) 23.25/23.49 = { by lemma 9 } 23.25/23.49 ifeq(p(i(i(i(i(X, Y), i(Z, Y)), W), i(i(Z, X), W))), true, ifeq(p(i(i(i(X, Y), i(Z, Y)), W)), true, p(i(i(Z, X), W)), true), true) 23.25/23.49 = { by axiom 4 (modus_ponens) } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 15: ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true) = true. 23.25/23.49 Proof: 23.25/23.49 ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true), true) 23.25/23.49 = { by lemma 14 } 23.25/23.49 ifeq(ifeq(p(i(i(i(i(Y, Z), i(W, Z)), i(X, i(W, Z))), i(i(W, Y), i(X, i(W, Z))))), true, p(i(i(X, i(Y, Z)), i(i(W, Y), i(X, i(W, Z))))), true), true, ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true), true) 23.25/23.49 = { by lemma 9 } 23.25/23.49 ifeq(ifeq(true, true, p(i(i(X, i(Y, Z)), i(i(W, Y), i(X, i(W, Z))))), true), true, ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true), true) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(p(i(i(X, i(Y, Z)), i(i(W, Y), i(X, i(W, Z))))), true, ifeq(p(i(X, i(Y, Z))), true, p(i(i(W, Y), i(X, i(W, Z)))), true), true) 23.25/23.49 = { by axiom 4 (modus_ponens) } 23.25/23.49 true 23.25/23.49 23.25/23.49 Lemma 16: p(i(i(i(i(X, Y), Y), Z), i(i(i(Y, X), X), Z))) = true. 23.25/23.49 Proof: 23.25/23.49 p(i(i(i(i(X, Y), Y), Z), i(i(i(Y, X), X), Z))) 23.25/23.49 = { by axiom 3 (ifeq_axiom) } 23.25/23.49 ifeq(true, true, p(i(i(i(i(X, Y), Y), Z), i(i(i(Y, X), X), Z))), true) 23.25/23.49 = { by axiom 2 (axL) } 23.25/23.49 ifeq(p(i(i(i(Y, X), X), i(i(X, Y), Y))), true, p(i(i(i(i(X, Y), Y), Z), i(i(i(Y, X), X), Z))), true) 23.25/23.49 = { by lemma 8 } 24.05/24.22 true 24.05/24.22 24.05/24.22 Lemma 17: p(i(X, i(Y, Y))) = true. 24.05/24.22 Proof: 24.05/24.22 p(i(X, i(Y, Y))) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(true, true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by lemma 14 } 24.05/24.22 ifeq(ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(ifeq(ifeq(true, true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by lemma 8 } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by lemma 11 } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 4 (modus_ponens) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 4 (modus_ponens) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y)))), i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by axiom 3 (ifeq_axiom) } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y)))), i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by lemma 16 } 24.05/24.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y)))), i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.22 = { by lemma 14 } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by lemma 13 } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(f(X, Y), Y), i(Y, Y))), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by lemma 14 } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 2 (axL) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))))), true, ifeq(p(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true), true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 4 (modus_ponens) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by lemma 14 } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by axiom 3 (ifeq_axiom) } 24.05/24.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.23 = { by lemma 14 } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by lemma 13 } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by lemma 8 } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, ifeq(true, true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 1 (axBp) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true, ifeq(p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 4 (modus_ponens) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(Y, Y), i(Y, Y)), i(i(i(Y, Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, p(i(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))), i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))))), true), true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by lemma 15 } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(Y, Y), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y))))), true, ifeq(p(i(i(Y, Y), i(Y, Y))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(i(Y, Y), i(Y, Y)))), true), true), true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 4 (modus_ponens) } 24.05/24.24 ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(p(i(i(i(i(Y, Y), i(Y, Y)), i(Y, Y)), i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y)))), true, p(i(i(i(f(X, Y), Y), i(Y, Y)), i(Y, Y))), true), true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by lemma 10 } 24.05/24.24 ifeq(ifeq(true, true, p(i(i(Y, f(X, Y)), i(Y, Y))), true), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(true, true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by lemma 8 } 24.05/24.24 ifeq(ifeq(p(i(X, i(Y, f(X, Y)))), true, p(i(i(i(Y, f(X, Y)), i(Y, Y)), i(X, i(Y, Y)))), true), true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(ifeq(ifeq(true, true, p(i(X, i(Y, f(X, Y)))), true), true, p(i(i(i(Y, f(X, Y)), i(Y, Y)), i(X, i(Y, Y)))), true), true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by lemma 11 } 24.05/24.24 ifeq(ifeq(ifeq(p(i(f(X, Y), f(X, Y))), true, p(i(X, i(Y, f(X, Y)))), true), true, p(i(i(i(Y, f(X, Y)), i(Y, Y)), i(X, i(Y, Y)))), true), true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by axiom 6 (resid1) } 24.05/24.24 ifeq(ifeq(true, true, p(i(i(i(Y, f(X, Y)), i(Y, Y)), i(X, i(Y, Y)))), true), true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(p(i(i(i(Y, f(X, Y)), i(Y, Y)), i(X, i(Y, Y)))), true, ifeq(p(i(i(Y, f(X, Y)), i(Y, Y))), true, p(i(X, i(Y, Y))), true), true) 24.05/24.24 = { by axiom 4 (modus_ponens) } 24.05/24.24 true 24.05/24.24 24.05/24.24 Lemma 18: p(i(i(X, Y), i(Z, i(X, Y)))) = true. 24.05/24.24 Proof: 24.05/24.24 p(i(i(X, Y), i(Z, i(X, Y)))) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(true, true, p(i(i(X, Y), i(Z, i(X, Y)))), true) 24.05/24.24 = { by lemma 17 } 24.05/24.24 ifeq(p(i(Z, i(Y, Y))), true, p(i(i(X, Y), i(Z, i(X, Y)))), true) 24.05/24.24 = { by lemma 15 } 24.05/24.24 true 24.05/24.24 24.05/24.24 Lemma 19: ifeq(p(i(i(i(i(X, Y), Y), i(i(Y, X), X)), Z)), true, p(Z), true) = true. 24.05/24.24 Proof: 24.05/24.24 ifeq(p(i(i(i(i(X, Y), Y), i(i(Y, X), X)), Z)), true, p(Z), true) 24.05/24.24 = { by axiom 3 (ifeq_axiom) } 24.05/24.24 ifeq(p(i(i(i(i(X, Y), Y), i(i(Y, X), X)), Z)), true, ifeq(true, true, p(Z), true), true) 24.05/24.24 = { by axiom 2 (axL) } 24.05/24.24 ifeq(p(i(i(i(i(X, Y), Y), i(i(Y, X), X)), Z)), true, ifeq(p(i(i(i(X, Y), Y), i(i(Y, X), X))), true, p(Z), true), true) 24.05/24.24 = { by axiom 4 (modus_ponens) } 24.29/24.48 true 24.29/24.48 24.29/24.48 Goal 1 (axK_axC): tuple(p(i(i(X, i(Y, Z)), i(Y, i(X, Z)))), p(i(c1, i(c2, c1)))) = tuple(true, true). 24.29/24.48 The goal is true when: 24.29/24.48 X = i(i(?, ?), ?) 24.29/24.48 Y = i(i(?, ?), ?) 24.29/24.48 Z = ? 24.29/24.48 where "?" stands for an arbitrary term of your choice. 24.29/24.48 24.29/24.48 Proof: 24.29/24.48 tuple(p(i(i(i(i(?, ?), ?), i(i(i(?, ?), ?), ?)), i(i(i(?, ?), ?), i(i(i(?, ?), ?), ?)))), p(i(c1, i(c2, c1)))) 24.29/24.48 = { by lemma 16 } 24.29/24.48 tuple(true, p(i(c1, i(c2, c1)))) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(true, true, p(i(c1, i(c2, c1))), true)) 24.29/24.48 = { by lemma 17 } 24.29/24.48 tuple(true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(true, true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 15 } 24.29/24.48 tuple(true, ifeq(ifeq(p(i(c1, i(i(c1, c1), c1))), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(true, true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 4 (modus_ponens) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))))), true, ifeq(p(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true), true), true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 18 } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true), true), true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(p(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true), true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 18 } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true), true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 4 (modus_ponens) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1))))), true, ifeq(p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 16 } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 14 } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by lemma 19 } 24.29/24.48 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.48 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by lemma 19 } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), i(f(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1))), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true, p(i(f(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1))), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by lemma 12 } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(f(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1))), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(f(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1))), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), c1), c1)), i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 6 (resid1) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(i(c1, c1), c1), c1), c1), i(i(c1, c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(c1, c1), i(i(i(c1, c1), c1), c1)), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by lemma 14 } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by lemma 13 } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(ifeq(p(i(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, ifeq(p(i(i(i(i(c1, c1), c1), c1), i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))))), true, p(i(i(i(i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1))), i(c1, i(i(c1, c1), c1)))), true), true), true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 4 (modus_ponens) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(ifeq(true, true, p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(ifeq(ifeq(p(i(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(c1, i(i(c1, c1), c1)))), true, ifeq(p(i(i(i(c1, i(i(c1, c1), c1)), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1)))), i(i(i(c1, c1), c1), i(c1, i(i(c1, c1), c1))))), true, p(i(c1, i(i(c1, c1), c1))), true), true), true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 4 (modus_ponens) } 24.29/24.49 tuple(true, ifeq(ifeq(true, true, p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 3 (ifeq_axiom) } 24.29/24.49 tuple(true, ifeq(p(i(i(c2, i(c1, c1)), i(c1, i(c2, c1)))), true, ifeq(p(i(c2, i(c1, c1))), true, p(i(c1, i(c2, c1))), true), true)) 24.29/24.49 = { by axiom 4 (modus_ponens) } 24.29/24.49 tuple(true, true) 24.29/24.49 % SZS output end Proof 24.29/24.49 24.29/24.49 RESULT: Unsatisfiable (the axioms are contradictory). 24.29/24.49 EOF