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Proof step 11 from proof no-progress-left-aux0 (previous step)

N = N'' + 1 until Boolvar, □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> G1(pm, S, S')] ∥ false ::- [PL S = s0]], \A ∀ N, Boolvar, R1, pm, S, R2, P, G1, Post2, s, G2, Pre1, Post1, I, Pre2. N < n → N = N'' +1 until Boolvar ∧ □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ (P(pm, S) ∨ last) ∧ [:S | Boolvar ::- [PL □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> G1(pm, S, S')] ∥ false ::- [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) ∧ stable(pm, I, R1) ∧ stable(pm, I, R2) ∧ I(pm, S) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) → false, n = N + 1, P(pm, s3), I(pm, S), R1(pm, s2, s3), G2(pm, s2, s3), P(pm, s2), R2(pm, s3, S), R1(pm, s3, S), P-restr(pm, P, I, R1), Pre2(pm, s), Pre1(pm, s), I(pm, s2), stable(pm, I, R2), stable(pm, I, R1), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), I(pm, s3), R1(pm, s2, S), P(pm, S), ¬ R2(pm, s3, s0) ⊦

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 12:
RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s1), I(pm, s0), □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s1, S);G1(pm, S, S'))] ∥ [PL S = s]], ¬ R2(pm, s0, s), R2(pm, s0, S) ⊦
Proof step 13:
N = N'' + 1 until Boolvar, □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> G1(pm, S, S')] ∥ false ::- [PL S = s0]], \A ∀ N, Boolvar, R1, pm, S, R2, P, G1, Post2, s, G2, Pre1, Post1, I, Pre2. N < n → N = N'' +1 until Boolvar ∧ □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ (P(pm, S) ∨ last) ∧ [:S | Boolvar ::- [PL □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> G1(pm, S, S')] ∥ false ::- [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) ∧ stable(pm, I, R1) ∧ stable(pm, I, R2) ∧ I(pm, S) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) → false, n = N + 1, P(pm, s3), I(pm, S), R1(pm, s2, s3), G2(pm, s2, s3), P(pm, s2), R2(pm, s3, S), R1(pm, s3, S), P-restr(pm, P, I, R1), Pre2(pm, s), Pre1(pm, s), I(pm, s2), stable(pm, I, R2), stable(pm, I, R1), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), I(pm, s3), R1(pm, s2, S), P(pm, S), ¬ R2(pm, s3, s0) ⊦ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ [PL S = s0]]

Rule argument

spec: RG-I-2-Theory inst: name: fair-never-violates-right-rely seq: RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s1), I(pm, s0), □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s1, S);G1(pm, S, S')) ] ∥ [PL S = s] ], ¬ R2(pm, s0, s), R2(pm, s0, S) ⊦ subst: [pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2, s1, s0, S, s] → [pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2, s, s3, S, s0] flags: []

Additional information

Used terms: pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2, s, s3, S, s0