[Imprint][Data Privacy Statement][E-Mail]

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Proof step 6 from proof rg2-sound-main-unfair (previous step)

( ( ( (((true ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R1(pm, s2, s2)) ∧ ¬ R2(pm, s3, S) ∨ (((true ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, S)) ∧ ¬ R1(pm, s2, s2)) ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))) ∨ (∃ s0. (¬ false ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ ¬ R1(pm, s3, s0)) ∧ ¬ boolvar ∧ [:S | [PL S = s0 ∧ true] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]]) ∨ (∃ s0, s1. ((((false ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R1(pm, s2, s0)) ∧ ¬ R2(pm, s3, s1)) ∧ [:S | [PL S = s0 ∧ true] ∥nf [PL S = s1 ∧ true]]) ∨ (∃ s0, s1. (((false ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, s1)) ∧ ¬ R1(pm, s2, s0)) ∧ [:S | [PL S = s0 ∧ true] ∥nf [PL S = s1 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]])) ∨ ( (((true ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ R1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R2(pm, s3, S) ∨ (((true ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ R1(pm, s2, s2)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, S)) ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))) ∨ (∃ s0. (¬ false ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ R1(pm, s3, s0)) ∧ ¬ boolvar ∧ [:S | [PL S = s0 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]) ∨ (∃ s0, s1. ((((false ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ R1(pm, s2, s0)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R2(pm, s3, s1)) ∧ [:S | [PL S = s0 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s1 ∧ true]]) ∨ (∃ s0, s1. (((false ⊃ Post1(pm, s, s2);G1(pm, s2, s2)) ∧ R1(pm, s2, s0)) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, s1)) ∧ [:S | [PL S = s0 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s1 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ])) ∨ ( ( (((true ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ R2(pm, s2, s2)) ∧ ¬ R1(pm, s3, S) ∨ (((true ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ R1(pm, s3, S)) ∧ ¬ R2(pm, s2, s2)) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))) ∨ (∃ s0. (¬ false ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ ¬ R2(pm, s3, s0)) ∧ ¬ boolvar ∧ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s0 ∧ true]]) ∨ (∃ s0, s1. ((((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ R2(pm, s2, s0)) ∧ ¬ R1(pm, s3, s1)) ∧ [:S | [PL S = s1 ∧ true] ∥nf [PL S = s0 ∧ true]]) ∨ (∃ s0, s1. (((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ R1(pm, s3, s1)) ∧ ¬ R2(pm, s2, s0)) ∧ [:S | [PL S = s1 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s0 ∧ true]])) ∨ ( (((true ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ R1(pm, s3, S) ∨ (((true ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R2(pm, s2, s2)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ R1(pm, s3, S)) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))) ∨ (∃ s0. (¬ false ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, s0)) ∧ ¬ boolvar ∧ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s0 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]) ∨ (∃ s0, s1. ((((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R2(pm, s2, s0)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ R1(pm, s3, s1)) ∧ [:S | [PL S = s1 ∧ true] ∥nf [PL S = s0 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]]) ∨ (∃ s0, s1. (((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R2(pm, s2, s0)) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3)) ∧ R1(pm, s3, s1)) ∧ [:S | [PL S = s1 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s0 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]), \A ∀ N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2. N < m → [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S) → ((R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), m = n, RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s2), I(pm, s), ¬ false, boolvar → s2 = s3 ∧ n = n0 ⊦ (false ⊃ Post1(pm, s, s2) ∧ Post2(pm, s, s2);G1(pm, s2, s3) ∨ G2(pm, s2, s3)) ∧ ( R1(pm, s3, S) ∧ R2(pm, s3, S) ∧ n = N + 1 → (R1(pm, S', S'') ∧ R2(pm, S', S'') ∧ N = N'' + 1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
Post1(pm, s, s2) ∧ G2(pm, s2, s3) ∧ ¬ R1(pm, s2, s2) ∧ ¬ R2(pm, s3, S) ∨ Post1(pm, s, s2) ∧ G2(pm, s2, s3) ∧ R2(pm, s3, S) ∧ ¬ R1(pm, s2, s2) ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S')) ∨ G1(pm, s2, s3) ∧ ¬ R1(pm, s3, s0) ∧ ¬ boolvar ∧ [:S | [PL S = s0] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∨ G1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ ¬ R1(pm, s2, s1) ∧ ¬ R2(pm, s3, s4) ∧ [:S | [PL S = s1] ∥nf [PL S = s4]] ∨ G1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ R2(pm, s3, s6) ∧ ¬ R1(pm, s2, s5) ∧ [:S | [PL S = s5] ∥nf [PL S = s6 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∨ Post1(pm, s, s2) ∧ R1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ ¬ R2(pm, s3, S) ∨ Post1(pm, s, s2) ∧ R1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ R2(pm, s3, S) ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S')) ∨ G1(pm, s2, s3) ∧ R1(pm, s3, s7) ∧ ¬ boolvar ∧ [:S | [PL S = s7 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∨ G1(pm, s2, s2) ∧ R1(pm, s2, s8) ∧ G2(pm, s2, s3) ∧ ¬ R2(pm, s3, s9) ∧ [:S | [PL S = s8 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s9]] ∨ G1(pm, s2, s2) ∧ R1(pm, s2, s10) ∧ G2(pm, s2, s3) ∧ R2(pm, s3, s11) ∧ [:S | [PL S = s10 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s11 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∨ Post2(pm, s, s2) ∧ G1(pm, s2, s3) ∧ ¬ R2(pm, s2, s2) ∧ ¬ R1(pm, s3, S) ∨ Post2(pm, s, s2) ∧ G1(pm, s2, s3) ∧ R1(pm, s3, S) ∧ ¬ R2(pm, s2, s2) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S')) ∨ G2(pm, s2, s3) ∧ ¬ R2(pm, s3, s12) ∧ ¬ boolvar ∧ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s12]] ∨ G2(pm, s2, s2) ∧ G1(pm, s2, s3) ∧ ¬ R2(pm, s2, s13) ∧ ¬ R1(pm, s3, s14) ∧ [:S | [PL S = s14] ∥nf [PL S = s13]] ∨ G2(pm, s2, s2) ∧ G1(pm, s2, s3) ∧ R1(pm, s3, s16) ∧ ¬ R2(pm, s2, s15) ∧ [:S | [PL S = s16 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s15]] ∨ Post2(pm, s, s2) ∧ R2(pm, s2, s2) ∧ G1(pm, s2, s3) ∧ ¬ R1(pm, s3, S) ∨ Post2(pm, s, s2) ∧ R2(pm, s2, s2) ∧ G1(pm, s2, s3) ∧ R1(pm, s3, S) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S')) ∨ G2(pm, s2, s3) ∧ R2(pm, s3, s17) ∧ ¬ boolvar ∧ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s17 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∨ G2(pm, s2, s2) ∧ R2(pm, s2, s18) ∧ G1(pm, s2, s3) ∧ ¬ R1(pm, s3, s19) ∧ [:S | [PL S = s19] ∥nf [PL S = s18 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∨ G2(pm, s2, s2) ∧ R2(pm, s2, s20) ∧ G1(pm, s2, s3) ∧ R1(pm, s3, s21) ∧ [:S | [PL S = s21 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s20 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], \A ∀ N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2. N < n → [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S) → ((R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s2), I(pm, s), boolvar → s2 = s3 ∧ n = n0 ⊦ (G1(pm, s2, s3) ∨ G2(pm, s2, s3)) ∧ ( R1(pm, s3, S) ∧ R2(pm, s3, S) ∧ n = N + 1 → (R1(pm, S', S'') ∧ R2(pm, S', S'') ∧ N = N'' + 1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')))

Rule argument

Additional information

Used simplifier rules: pre2-i: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre2(pm, s) → I(pm, s);