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

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Proof step 7 from proof rg2-sound-main-unfair (previous step)

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')))

User applied rule case distinction

And obtained the following new goals

Proof step 8:
[:S | [PL S = s0] ∥nf [PL 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), ¬ R1(pm, s3, s0), ¬ boolvar ⊦ (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')))
Proof step 12:
[:S | [PL S = s1] ∥nf [PL S = s4]], \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, s2), G2(pm, s2, s3), ¬ R1(pm, s2, s1), ¬ R2(pm, s3, s4) ⊦ (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')))
Proof step 14:
[:S | [PL S = s5] ∥nf [PL S = s6 ∧ 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, s2), G2(pm, s2, s3), R2(pm, s3, s6), ¬ R1(pm, s2, s5) ⊦ (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')))
Proof step 19:
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, Post1(pm, s, s2), R1(pm, s2, s2), G2(pm, s2, s3), R2(pm, s3, S) ⊦ (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')))
Proof step 25:
[: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'))]], \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), R1(pm, s3, s7), ¬ boolvar ⊦ (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')))
Proof step 29:
[:S | [PL S = s8 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s9]], \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, s2), R1(pm, s2, s8), G2(pm, s2, s3), ¬ R2(pm, s3, s9) ⊦ (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')))
Proof step 34:
[: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'))] ], \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, s2), R1(pm, s2, s10), G2(pm, s2, s3), R2(pm, s3, s11) ⊦ (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')))
Proof step 38:
[:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s12]], \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, G2(pm, s2, s3), ¬ R2(pm, s3, s12), ¬ boolvar ⊦ (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')))
Proof step 42:
[:S | [PL S = s14] ∥nf [PL S = s13]], \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, G2(pm, s2, s2), G1(pm, s2, s3), ¬ R2(pm, s2, s13), ¬ R1(pm, s3, s14) ⊦ (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')))
Proof step 44:
[:S | [PL S = s16 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s15]], \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, G2(pm, s2, s2), G1(pm, s2, s3), R1(pm, s3, s16), ¬ R2(pm, s2, s15) ⊦ (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')))
Proof step 49:
R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(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, Post2(pm, s, s2), R2(pm, s2, s2), G1(pm, s2, s3), R1(pm, s3, S) ⊦ (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')))
Proof step 55:
[: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'))]], \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, G2(pm, s2, s3), R2(pm, s3, s17), ¬ boolvar ⊦ (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')))
Proof step 59:
[:S | [PL S = s19] ∥nf [PL S = s18 ∧ 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, G2(pm, s2, s2), R2(pm, s2, s18), G1(pm, s2, s3), ¬ R1(pm, s3, s19) ⊦ (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')))
Proof step 64:
[: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, G2(pm, s2, s2), R2(pm, s2, s20), G1(pm, s2, s3), R1(pm, s3, s21) ⊦ (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

pos: left 1

Additional information

Used simplifier rules: r1-refl: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ ¬ R1(pm, s, s) ∧ I(pm, s) → false;%g2-r1: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ G2(pm, s, s0) ∧ I(pm, s) → R1(pm, s, s0);%r1-refl: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ ¬ R1(pm, s, s) ∧ I(pm, s) → false;%g2-r1: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ G2(pm, s, s0) ∧ I(pm, s) → R1(pm, s, s0);%r2-refl: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ ¬ R2(pm, s, s) ∧ I(pm, s) → false;%g1-r2: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ G1(pm, s, s0) ∧ I(pm, s) → R2(pm, s, s0);%r2-refl: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ ¬ R2(pm, s, s) ∧ I(pm, s) → false;%g1-r2: ⊦ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ G1(pm, s, s0) ∧ I(pm, s) → R2(pm, s, s0);