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

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Proof step 16 from proof fair-sustains-implies-always (previous step)

boolvar ∨ n = N + 1 ∧ N = N'' + 1 until Boolvar, □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), ( ( (((R1(pm, s2, s2) ∧ (true ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R2(pm, s3, S) ∨ false) ∨ ((((R1(pm, s2, s2) ∧ (true ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ boolvar) ∧ ¬ R2(pm, s3, S)) ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∨ (((R1(pm, s2, s2) ∧ (true ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, S)) ∧ ¬ boolvar) ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))) ∨ (∃ s0. (¬ boolvar ∧ R1(pm, s3, s0) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL S = s0 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]) ∨ ( (∃ s0, s1. (((R1(pm, s2, s0) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ R2(pm, s3, s1)) ∧ [:S | false ::- [PL S = s0 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s1 ∧ true]]) ∨ (∃ s0. false)) ∨ (∃ s0, s1. ((((R1(pm, s2, s0) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3))) ∧ ¬ boolvar) ∧ ¬ R2(pm, s3, s1)) ∧ [:S | false ::- [PL S = s0 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s1 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')] ]) ∨ (∃ s0, s1. (((R1(pm, s2, s0) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s2))) ∧ (false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, s1)) ∧ ¬ boolvar) ∧ [:S | false ::- [PL S = s0 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s1 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ])) ∨ ( ( (((true ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R1(pm, s3, S) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ 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)) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ true]]) ∨ (∃ s0, s1. (((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R1(pm, s3, s1) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ R2(pm, s2, s0)) ∧ [:S | false ::- [PL S = s1 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ true]])) ∨ false ∨ (¬ false ∧ false) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL true ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]) ∨ ( (∃ s0. (¬ false ∧ ((false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ ¬ boolvar) ∧ ¬ R2(pm, s3, s0)) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')] ]) ∨ (∃ s0, s1. ((((false ⊃ Post2(pm, s, s2);G2(pm, s2, s2)) ∧ R1(pm, s3, s1) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ boolvar) ∧ ¬ R2(pm, s2, s0)) ∧ [:S | false ::- [PL S = s1 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')] ])) ∨ (∃ s0. (¬ false ∧ ((false ⊃ Post2(pm, s, s2);G2(pm, s2, s3)) ∧ R2(pm, s3, s0)) ∧ ¬ boolvar) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ 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)) ∧ R1(pm, s3, s1) ∧ (false ⊃ Post1(pm, s, s2);G1(pm, s2, s3))) ∧ ¬ boolvar) ∧ [:S | false ::- [PL S = s1 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s0 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ]), \A ∀ N, R1, pm, S, R2, Boolvar, Post1, s, G1, Post2, G2, Pre1, I, Pre2. N < m → □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL ¬ Boolvar until ¬ R2(pm, S', S'') ∧ 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) ∧ I(pm, s) → false, 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, boolvar1 → s2 = s3 ∧ (boolvar ↔ boolvar0) ∧ n = n0, R1(pm, s3, S), R2(pm, s3, S) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
boolvar ∨ n = N + 1 ∧ N = N'' + 1 until Boolvar, □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), ¬ boolvar ∧ R1(pm, s3, s0) ∧ G1(pm, s2, s3) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL S = s0 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∨ R1(pm, s2, s1) ∧ G1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ ¬ R2(pm, s3, s4) ∧ [:S | false ::- [PL S = s1 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s4]] ∨ R1(pm, s2, s5) ∧ G1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ ¬ boolvar ∧ ¬ R2(pm, s3, s6) ∧ [:S | false ::- [PL S = s5 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s6 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')] ] ∨ R1(pm, s2, s7) ∧ G1(pm, s2, s2) ∧ G2(pm, s2, s3) ∧ R2(pm, s3, s8) ∧ ¬ boolvar ∧ [:S | false ::- [PL S = s7 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s8 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ 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) ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S')) ∨ G2(pm, s2, s3) ∧ ¬ R2(pm, s3, s9) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s9]] ∨ G2(pm, s2, s2) ∧ R1(pm, s3, s11) ∧ G1(pm, s2, s3) ∧ ¬ R2(pm, s2, s10) ∧ [:S | false ::- [PL S = s11 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s10]] ∨ G2(pm, s2, s3) ∧ ¬ boolvar ∧ ¬ R2(pm, s3, s12) ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s12 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')]] ∨ G2(pm, s2, s2) ∧ R1(pm, s3, s14) ∧ G1(pm, s2, s3) ∧ ¬ boolvar ∧ ¬ R2(pm, s2, s13) ∧ [:S | false ::- [PL S = s14 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s13 ∧ ¬ Boolvar until ¬ R2(pm, S', S'')] ] ∨ G2(pm, s2, s3) ∧ R2(pm, s3, s15) ∧ ¬ boolvar ∧ ¬ boolvar1 ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s15 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∨ G2(pm, s2, s2) ∧ R2(pm, s2, s16) ∧ R1(pm, s3, s17) ∧ G1(pm, s2, s3) ∧ ¬ boolvar ∧ [:S | false ::- [PL S = s17 ∧ □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL S = s16 ∧ ¬ Boolvar until ¬ R2(pm, S', S'') ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], \A ∀ N, R1, pm, S, R2, Boolvar, Post1, s, G1, Post2, G2, Pre1, I, Pre2. N < n → □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | false ::- [PL □ R1(pm, S', S'') ∧ □ (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ Boolvar ::- [PL ¬ Boolvar until ¬ R2(pm, S', S'') ∧ 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) ∧ I(pm, s) → false, RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s2), R1(pm, s3, S), R2(pm, s3, S), I(pm, s), boolvar1 → s2 = s3 ∧ (boolvar ↔ boolvar0) ∧ n = n0 ⊦

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);