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Proof step 14 from proof progress-left (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 15:
RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre2(pm, s1), I(pm, s0), □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | [PL S = s] ∥ [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s1, S);G2(pm, S, S'))]], ¬ R1(pm, s0, s), R1(pm, s0, S) ⊦
Proof step 16:
□ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | [PL S = s0 ∧ □ ◇ P(pm, S)] ∥ [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]], ¬ P(pm, s2) ∧ ¬ ◇ P(pm, S) ∨ n = N + 1 ∧ N = N'' + 1 until ¬ ◇ P(pm, S), \A ∀ N, P, pm, S, Post1, s, G1, R1, Post2, G2, R2, Pre1, I, Pre2. N < n → [:S | [PL □ ◇ P(pm, S) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ] ∧ □ (R1(pm, S', S'') ∧ R2(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) → ¬ N = N'' +1 until ¬ ◇ P(pm, S), R2(pm, s2, S), I(pm, s3), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), stable(pm, I, R1), stable(pm, I, R2), I(pm, s2), Pre1(pm, s), Pre2(pm, s), P-restr(pm, P, I, R1), R1(pm, s3, S), R2(pm, s3, S), G1(pm, s2, s3), R2(pm, s2, s3), I(pm, S), ¬ R1(pm, s3, s0) ⊦ [:S | [PL S = s0] ∥ [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]]

Rule argument

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